The only ancient text which mentions the parents of the Stymphalian birds makes it sound like these killer avians were Arkadian [Arcadian] princesses.
A scholion on Apollonius Rhodius' Argonautika 2.1054 preserves a statement made by the writer Mnaseas of Patrae. As with other things that Mnaseas has to say, it isn't quite the interpretation you might expect. William Smith's Dictionary of Greek and Roman Biography and Mythology summarises the information from that Argonautika scholion by saying that according to Mnaseas, the Stymphalides, as these monsters are also called,
were not birds, but women, and daughters of Stymphalus and Ornis, and
were killed by Heracles because they did not receive him hospitably.
Herakles [Heracles] was especially peeved about this because the Stymphalides had offered similar hospitality to Eurytos [Eurytus] and Kteatos [Cteatus], the twin army-generals of King Augeias of the Epeians of Elis. Eurytos and Kteatos are typically known as the Molionidai [Molionidae or Molionids], are sometimes described as gigantic or monstrous, and, being stronger than Herakles, they were the only men he was never able to defeat in battle. Herakles eventually bested them by cheating: he waylaid them during a sacred ceasefire, thus managing to kill them.
Pausanias' opinion about the birds, laced with some slightly jingoistic ethnicism, and expressed in Book 8 of his Description of Greece, is that perhaps the Stymphalides are merely naturally occurring animals, simply like any other bird type, these ones in particular having migrated into Greece from "the Arabian desert".
Whether the modern Arabian birds with the same name as the old
Arkadian birds are also of the same breed, I do not know.
But if there have been from all time Stymphalian birds, just as there
have been hawks and eagles, I should call these birds of Arabian
origin, and a section of them might have flown on some occasion to
Arkadia [Arcadia] and reached Stymphalos [Stymphalus]. Originally they would be
called by the Arabians, not Stymphalian, but by another name. But the
fame of Herakles [Heracles], and the superiority of the Greek over the
foreigner, has resulted in the birds of the Arabian desert being
called Stymphalian even in modern times.
There is also some enigmatic information from the temple of Artemis Stymphalia (Artemis of Stymphalos) which makes the Stymphalides sound like Harpies.
In Stymphalos there is also an old sanctuary of Artemis Stymphalia...
Near the roof of the temple have been carved, among other things, the
Stymphalian birds... There are here also maidens of white marble, with
the legs of birds, and they stand behind the temple.
The place being called Stymphalos here is a town in Arkadia which was named after King Stymphalos of Arkadia, who reigned in the time of Herakles' great-grandfather Pelops. The lake which was inhabited by the monstrous birds in question was in or near Stymphalos Town.
King Stymphalos, who was killed by Pelops, was a grandson of Arkas [Arcas], the son of Zeus from whom Arkadia received its name. Stymphalos's cousin Aleos [Aleus], who had reigned in conjunction with him, was still alive in Herakles' time (Herakles consorted with a daughter of Aleos and participated together with Aleos' sons in the Argonautic expedition).
Ornis is an interestingly mysterious character. She does not appear to be the mother of Stymphalos's other children, who are all apparently regular humans. Her name simply means "Bird." Perhaps she is some sort of part-bird monster nymph like the Seirenes [Sirens]. She is otherwise completely unknown.
Apollodorus' Bibliotheka tell us that King Stymphalos had a daughter named Parthenope, who bore Herakles a son called Everes. Incidentally, in Strabo and Lycophron, Parthenope is the name of one of the Seirenes, probably unrelated to the Arkadian princess, but all the same a curious coincidence (I think) considering the bird-legged ladies at the Artemis temple in Stymphalos.