Paris of Troy was presumably an adult at the time of the marriage of Peleus and Thetis and the Judgement of Paris. That would mean Paris was at least 15-20 years older than Achilles. Other than that, I have no idea how old either was at the time of the Trojan War.
Is there any mention in the Iliad, or any other ancient source of their ages?
Time and chronology tend to be fairly loose in Greek mythology, particularly as there are usually multiple authors treating the same subjects.
Regardless, Achilles is presumed to be a youth at the start of the war. In this famous vase painting, he is depicted as beardless:
You can read more about this vase at the Perseus site at Tufts.
You may recall that Cúchulainn, a warrior of similar prowess and invincibility, is also noted to be beardless during the action of the Táin, which causes some warriors to balk at dueling him. In this passage he resorts to using berry juice and grass to fake a beard. [The Tain, Kinsella, p.134]
Both Achilles and Cúchulainn performed amazing martial feats in childhood--the latter killed the monstrous hound from which his name derives, and the former was said to have killed his first boar by age 6. Thus extreme youth, as early as 11 or 12, is not implausible for Achilles at the outset of the war.
There is also the very well known story from several post-Homeric sources of Thetis spiriting Achilles away to Skyros to dress as a girl and hide among the women there as a way of avoiding his tragic fate were he to sail to Troy. When Nestor, Ajax and Odysseus went to suss him out, it is mentioned they never would have pierced the disguise had Odysseus not tricked him into revealing himself. (Legend holds that Odysseus dumped out a pile of gifts for the girls, but also included a shield and spear, which Achilles seized, stripping himself to the waist. [The Greek Myths, Graves, 160k]
This implies he has to be young enough to pass for a girl. Unless Achilles was extremely feminine, a possible implication is that his voice had not yet changed.
Achilles was also said to have fathered Neoptolemus at Skyros, but I'd caution against imposing contemporary mores in terms of appropriate ages for fatherhood. He could have just hit puberty when he conceives his son.
Robert Graves mentions the age of 15, citing Homer himself [The Greek Myths, Graves, 160.l], but I haven't been able to track down the exact passage—the version of the Iliad Graves was working from may have had different numbering or may have had passages that are now considered apocryphal. However, he does lead to some useful passages. In one, Phoenix says:
It was to thee that the old horseman Peleus sent me on the day when he sent thee to Agamemnon, forth from Phthia, [440] a mere child, knowing naught as yet of evil war...
What is interesting about that passage, when you go to Greek, is that Homer uses the term νήπιον ("nhay-pee-on") which not only means child, but in some cases, specifically a pre-pubescent child. (The LSJ entry from the word link makes direct reference to this passage in the Iliad.)
My best guess is that he would have been in his mid-teens at the outset of the war, but he certainly could have been a few years younger or older.
As for the age of Paris, that's much more of a rabbit hole. However, based on the age differential you mention, we can infer that he was in his late twenties to his mid-thirties.
Hope this answer provides some satisfaction. I'll continue to try and hunt down the mystery passage cited by Graves.
The Actual Duration of the War
Towards the end of the Iliad, whose action takes place in the tenth and final year of the Trojan War, Helen says (in Line 765 of Book 24) that "this is now the twentieth year" since she departed from Lakedaimon [Lacedaemon] with Paris. "The words have puzzled the Scholiasts and commentators"1 for centuries because one might have expected the duration mentioned by her to be closer to the length of the war; but the war's length accounts for only half of the amount of time she's talking about.
Having been lost in ancient times, the portion of the work attributed to Apollodorus entitled the Epitome was discovered in 1891, and it contains a full explanation for this duration. For the Greeks the war had indeed lasted about twenty years. They had landed in Troas only nine years prior to the beginning of the Iliad and had engaged the Trojans immediately. However, they had first mustered up their forces to head for their enemy's territory in Asia eight years before their arrival in Troas.
Not really knowing their way there, combined with additional shenanigans that transpired, they landed in Mysia, mistakenly thinking that that was Troas, and eventually all ended up back home in Greece. They only managed to reassemble eight years after that debacle, which is narrated in Epitome 3.17-19ff. Prior to the first assembly, they had spent two years making preparations to build their army. So 2 years of prep + 8 years of reorientation & re-prep + 10 years of combat = Helen's 20 years as mentioned in the Iliad.
Akhilleus' [Achilles'] Age
Just before the chaos at the first gathering ensues, the Epitome tells us (3.16):
So Agamemnon in person was in command of the whole army, and Akhilleus was admiral, being fifteen years old.
[Emphasis mine]
The Iliad mentions nothing so specific about the ages of Akhilleus and Paris, but the Epitome does inform us of how old Akhilleus is when the timeline is set at eight years before the beginning of the siege of Troy.
15 + 8 = 23. So Akhilleus is about twenty-three years old when the Greek army lands in Troas. Akhilleus dies in the last year of the war a decade later = about Age 33.
This also accounts for the pesky issue of the fact that Akhilleus' son Neoptolemos [Neoptolemus] arrives at Troy in the same year that the city falls, by then evidently a full-grown warrior of a man! If Neoptolemos was born in the same year of the first army gathering, or even the year after that (allowing that he could've been born in his father's absence), he would be roughly 17 or 18 years old when he helps to take Troy.
Paris's Age
Like Akhilleus, Paris, when he was quite young, had a son. Paris's childhood sweetheart was a Naiad named Oinone [Oenone]. His intrigue with her does not receive mention in the Iliad, but other mythographers tell us that Paris was already married to Oinone when he met Helen, and even before he met Hera, Athena and Aphrodite.
The son of Paris and Oinone was called Korythos [Corythus], who, together with his mother, was abandoned by Paris when the Trojan prince left Asia in pursuit of Helen. After Paris had returned to Troy with the Lakedaimonian [Lacedaemonian] queen, as we know, the Greek army was eventually coming for him.
As observed above, this army did not enjoy immediate success finding its way to Troy. Oinone, perhaps because she was a prophetess (as in Apollodorus' Bibliotheka and Ovid's Heroides), was apparently aware of this since she sent Korythos to the Greeks to help them navigate (going by Lycophron's Alexandra 57 [via Tzetzes' commentary]).
In Conon's Narrations 23, Oinone sends Korythos to Helen, while in Parthenius' Erotika Pathemata 34, Korythos
came to Troy to help the Trojans, and there fell in love with Helen. She indeed received him with the greatest warmth—he was of extreme beauty—but his father discovered his aims and killed him.
By all accounts Korythos was at least a young man by the time of the war, and if we follow Lycophron on this, we was old enough to set out by himself to go meet a fleet of enemy ships and offer them directions which they, presumably, received well enough.
I would say that, reasonably, he should have been at least fifteen years old at that point. This would be, perhaps, roundabout the year before the siege of Troy begins. (I.e. Korythos guided the Greeks to Troy and they attacked in earnest when they landed the following year.) If we borrow the age calculations of Akhilleus from the breakdown above, let's say that Paris was also at least fifteen years old when he had Korythos.
15 + 15 = 30. Paris is (at least) thirty years old the year before the Greeks make landfall in Troas.
30 + 10 = 40. Paris dies at around the age of forty in the last year of the Trojan War.
There isn't much difference in the final result if Paris is 20 at the birth of Korythos and/or if Korythos is five years younger than in the preceding calculation (if we allow that Korythos can be younger in the versions of his story which don't involve him granting help to the Greek military). In the higher estimate, Paris would be about 35 at the beginning of the siege, and 45 at death.2
1. Footnote 4 on p. 187 of Apollodorus: The Library, with an English Translation, in Two Volumes by Sir James George Frazer. 1921. William Heinemann, London, & G.P. Putnam's Sons, New York
2. Assuming that Akhilleus was born in the same year as his parents' wedding; and allowing that Paris was about, say, eighteen when he adjudicated the beauty contest of the goddesses, this would mean that the goddesses had been squabbling for at least six years, between that inauspicious wedding and their fateful meeting with the Trojan prince. The calculations for this are as follows:
45 - 33 = 12. Paris is (at most) 12 years older than
Akhilleus (based on their ages at death).18 - 12 = 6 years between the wedding of Peleus and Thetis & the Judgement of Paris.I don't believe we know when the Judgement of Paris happened in relation to the Wedding of Peleus and Thetis. It's usually left vague. I suppose we can just chalk this up to human years not feeling the same for the gods.
The Bibliotheca Epitome says Achilles is 15 at the first gathering in Aulis, which is 2 years after "[Paris] persuaded Helen to go off with him." However, since this text goes with the 20 year timeline, the Achaeans raid the wrong city and have to wait another 8 years. So, at the actual beginning of the war, Achilles should have been 23.
Philostratus' Heroica says Paris was 18 when he travelled to Greece to "seize Helen." So, by the 20 year timeline, Paris would have been 28 at the outbreak of war.
Another way to deduce the age of Paris is from the story that Paris killed his son Korythos when he found Korythos and Helen together. This implies that Paris was jealous because he consider Korythos old enough to be sexy.
So if Korythos was at least 14 years. old, and if Paris was at least 14 years older than his son Korythos, he should have been at least 28 during that incident, which could have been years before the end of the Trojan War.
Or assume that Korythos was about 20 to 30 years old at the time and that Paris was about 20 to 30 years older than his son Korythos. Thus Paris should have been 40 to 60 years old when he killed Korythos according to that story.
Adinkra's answer based on the age of his son Korythos calculates that Paris was about 30 when the Greeks landed at Troy and 40 when he died.
Adinkra's answer also calculates that Achilles should have been about 23 when the Greeks landed at Troy and about 33 when he died in the last year of the Trojan War.
Assuming that Paris was a young man about 20 to 30 years old at the time of the Judgement of Paris, as he is usually depicted, And if Achilles was born after the Judgement of Paris, and if Achilles was 33 years old when he died as in Adinkra's answer, Paris should have been about 53 to 63 years old when he killed Achilles, soon before Paris died.
And the problem is that Paris is usually depicted as a young man about 20 to 30 years old, often in depictions of events in the last year of the Trojan War, such as scenes from the Iliad, or the death of Achilles.
And that tends to imply that the age of Paris should have been a negative number when he ran off with Helen or when he made the Judgement of Paris!
So possibly some of the data used to estimate the age of Paris come from stories that are not part of the main myth cycle and can be disregarded.
Even if one assumes that Achilles was born years before the Judgement of Paris, the gods taking many years after the marriage of Achilles's parents to find a naive mortal to trick into making the Judgement of Paris, the Judgement of Paris still has to be before Paris and Helen run off to troy, which has to be before the Greeks begin the Trojan War twenty years before the death of Paris and the Fall of Troy.
So if Paris was 20 to 30 years old at his death in the last year of the Trojan war, he should have been aged 0 to 10 years old at the time of the Judgement of Paris and the time he ran off with Helen to Troy.
So I think that it would make chronological sense to depict Paris as a naive child when he is suckered into making the Judgement of Paris, and Paris and Helen as gullible children or at most teenagers when they follow the advice of the Goddess Aphrodite and her assurances that everything will work out all right ("after all, what's the worst thing that could possibly happen") and run away to Troy.
Achilles being in his early 20s makes sense, he's around 10 when Helen is brought to Troy and he goes into hiding, and is there at least until his early teens since he fathers Neoptolemus there.
Given that Paris is already married to Oenone at the time of the Judgement he's likely at least in his late teens when Peleus and Thetis marry. Even if Achilles is born within a year of the wedding (unlikely), Paris is still at least 15 years older than Achilles, probably closer to 20. That puts Paris in his late 30s when the Greeks arrive, and late 40s at his death. He is most definitely older than Achilles.
