I don't feel able to help you out with Plato and the Bhagavad Gita, but Parmenides is different from the others. There it's more a poetic trope than a philosophical concept: remember, (a) the narrator isn't driving the chariot himself: it's the (divine) maidens who are leading him on his journey. And (b) it's probably obvious to you already, but it's always worth a reminder, that Parmenides didn't see himself as a forerunner to Plato: he was a poet working within a tradition of wisdom literature like the poetry of Theognis and Hesiod. His only potential model on the philosophical side of things was Xenophanes, and I hope it's reasonably obvious that Parmenides is much more literary than Xenophanes.

Later poets -- and I mean purely poetic poets, not philosophers -- borrowed from Parmenides too. Take this passage which probably comes from the proem of Choirilos' Persika (an epic about the Persian Wars):

Ah! blessed he who was skilled in song at that time,
a servant of the Muses when the meadow was still undefiled.
But now, everything’s been parcelled out, the arts have their endings;
we’re left behind, as the last in the race. There’s no way,
anywhere one looks, to bring up a new-yoked chariot.

I'd interpret the chariot in Parmenides as part of a set of reversals of conventional tropes throughout the poem's proem. He's not driving the chariot, he's being led; the poem is a piece of wisdom literature, but the narrator isn't the one doing the teaching, he's being taught. The latter is a pretty clear allusion to the opening of the Hesiodic Theogony, where Hesiod is instructed by the Muses. This helps frame the content of the poem, building the opposition of what is and is-not on top of the poetic trope of two kinds of knowledge (like Hesiod's Muses who know how to tell true things, and also things that sound like the truth but are not). The visitation to the goddess may (possibly; I'm not so confident about this) may be an allusion to the Derveni Theogony, or another poem like it, where Zeus goes to the cave of Night to be instructed in the nature of the universe. If that's right, then that suggests he's framing the poem as a piece of hierophantic quasi-religious instruction. How exactly the chariot fits into that isn't very clear. But I think literary models are the kind of source we should be thinking of.

Obviously none of this applies to Plato's charioteer!