As with anything in Hinduism, there is an overabundance of sources rather than a lack of one.
Taking examples from just one text; Abirami Andhadhi, a poem on Abirami which is another name for Parvati (or Gauri, the consort of Shiva as mentioned in the other answer):
The very first stanza of the poem describes the various similes to the red colour of the Goddess: the rising sun in the morning (உதிக்கின்ற செங்கதிர்), the red kumkum kept on the forehead (உச்சித்திலகம்), a ruby stone (மாணிக்கம்), the flower of pomegranate (மாதுளம்போது), all these are given as examples of how her skin looks.
But then, a few verses later (v:15), the poem compares her to a "green parrot" (பைங்கிளியே ), and then later (v:70) calls her someone who appears in a "green that denotes a joyful land" (மண் களிக்கும் பச்சை வண்ணமும் ஆகி ... தோன்றிய எம்பெருமாட்டி).
The best of all is verse 21 though, which calls her "பிங்கலை நீலி செய்யாள் வெளியாள் பசும்பெண்கொடி":
* பிங்கலை refers to a yellow or golden-skinned person
* நீலி directly translates to "blue-coloured"
* செய்யாள் translates to "red-coloured"
* வெளியாள் translates to "white-coloured"
* பசும் பெண் கொடி calls her "green coloured
All within a single line!
And this poem is not a special case either: most of these references go back to older texts (eg. Lalitha Sahasra Nama or Sankara's Soundarya Lahari), which too assign varying colours to the Goddess.
And this isn't limited to Goddess Parvati too. Shiva gets white, black or blue skin in different places, Vishnu goes through black, blue, red and green in different texts, and so on. One of Vishnu's avatars, Krishna, actually directly translates to "the dark-skinned one", but even he usually gets blue-skin in illustrations, as the question mentions. (My theory is that this is actually linguistic misunderstanding: the original Sanskrit calls him "neela" coloured, which back then refered to any dark colour, black or dark blue or dark green; by the time of these illustrations though, the common languages had "neela" meaning just blue.)
The reasons for this can be given from two viewpoints:
from a historical view, the reason there's so much variation is (a) these texts were generally written by poets in the Bhakthi tradition, which is all about emotion and personal connection with God rather than analytical accuracy (b) Hinduism has absorbed various local beliefs through time, and allowed them all place within itself, turning those regional Gods into forms of the original Hindu Gods (so for eg. when Maariyamma மாரியம்மா a Tamil regional Goddess was taken in as a form of Goddess Parvati, Parvati acquired her dark skin as one of her colours).
from a philosophical view, it is a constant reminder to not get attached to the form of the God, that names and forms (Naama-Roopa) - even those of Gods - are only products of Maya, and the Truth is beyond such limitations.