I'm not sure what motivated Parmenides because he was more of a poet than anything - it just happened that his poetry was what we would now recognize as incredibly philosophical. He didn't really argue, he just wrote down what the "goddess" told him. But I think the basic problem is that everyone back then agreed that you can't get "something from nothing," and it sure seems like change requires being to come from non-being. The statue is there now, but before it was cast there wasn't a statue, just a chunk of bronze. If being can't come from non-being, how do you account for the "coming-to-be" of the statue? The Eliatic position as I understand it is that the change is just an illusion. Plato and Aristotle both react against this position and argue that it's silly (I'm very inclined to agree). They then give alternative accounts of what change really is.

I'm not sure about Plato, but the Aristotelian analysis is something like this: every thing that exists has the potential to exist in certain ways and not others, and it's said that the thing is "in potency" to exist in those potential ways. When something could exist in a certain way but right now doesn't, that's called a "privation." And the ways that the thing currently does exist are the "form" of the thing. So a substance changes when it goes from being in potency to being actual, and it does that by losing a privation. Aquinas follows Aristotle in giving the example: "For example, when a statue is made from bronze, the bronze which is in potency to the form of the statue is the matter; the shapeless or undisposed something is the privation; and the shape because of which it is called a statue is the form." Incidentally, Aquinas's short On the Principles of Nature (https://aquinas.cc/la/en/~DePrinNat) is a good overview of this theory, which is spread all over Aristotle (in the Categories, the Physics, and the Metaphysics).

As far as οὐσία is concerned, I think this is the complete Greek for Parmenides's poem: http://philoctetes.free.fr/parmenidesunicode.htm. In the places where that translation uses "being" you get slightly different words like γενέσθαι (to come into a new state of being) or εἶναι (just the infinitive "to be"). And looking at the definition of οὐσία (https://lsj.gr/wiki/%CE%BF%E1%BD%90%CF%83%CE%AF%CE%B1) it looks like most of the uses of that term specifically come well after Parmenides.