It is curious that Parmenides's argument seems wrong today while back in the day someone as expert as Plato thought that it was obviously right. Indeed, one of the drives behind Plato's system was resolving the puzzle that Parmenides and Heraclitus were both right. Let me strip the argument down to bare bones.
P1(Panlogicism) Only that which is thinkable is (as Plato writes in Parmenides, "...for the same thing can be thought and can exist").
P2(Law of Identity) Only self-identical is thinkable (in Aristotle's retelling:"Parmenides of Elea, improving upon the ruder conceptions of Xenophanes, was the first to give emphatic proclamation to the celebrated Eleatic doctrine... i.e. the Cogitable... enduring and unchangeable, of which the negative was unmeaning, and the Sensible or Perceivable").
P3(Many is not One) Change/becoming is not self-identical.
C1) Change/becoming is not thinkable (from P2 and P3).
C2) Change/becoming is not (from P1 and C1).
The argument is obviously valid, the question can only be if it is sound. If the conclusions are undesirable one has to take issue with the premises. P3 seems almost tautological, but according to Papa-Grimaldi's Why Mathematical Solutions of Zeno’s Paradoxes Miss The Point Zeno's paradoxes were directed against the "Pythagorean pretence" of spinning the many out of the one nonetheless, the motion out of the rest, the change out of the self-identity. What of our experience? Well, Parmenides and Zeno did not deny that becoming is experiencable, but that does not mean that it is, or that it is cogitable/conceptualizable by our thought. Hence Zeno's (and Plato's) contention that change is an illusion. As it applies to flying arrow:
"A manipulation of this unit like the one accomplished by the Pythagoreans will not give us a plurality, a real dynamism,
but simply a repetition of the identical unit. That is, many other selfidentical positions in which the arrow is found “at rest”. But how the transition from one position to the next has been accomplished remains for our thought a mystery."
Aristotle affirmed the law of identity, logic was needed to make the science possible, even if it is a science of becoming, but denied the panlogicism. In Felt's words from Impossible Worlds:
"The link between Actuality and Possibility lies not in possibilities but in potentiality... Thus the new actual is always growing out of the womb of the potential, but the potential is itself rooted in and structured by past actuality. The actualists are therefore right in denying an independence to the possible. On the other hand, to be potentially is really a way to be, even though it is not to be actually. And this of course is just what Aristotle said in response to Parmenides, who conceived of only one way of being, being in actuality."
Hegel, on the other hand, kept the panlogicism, but denied the law of identity, along with the law of non-contradiction, in conceiving his dialectic, the "logic" of becoming. As Papa-Grimaldi puts it:
"...we should Hegelianly rise above the “thinking that belongs to the understanding alone” and have an intuition of the arrow as never occupying a space equal to itself. This is the Hegelian key to the interpretation of reality and movement: to deny the identity as a constraint on our reasoning and rather opt for the speculative Reason that raises itself above “the mere logic of the understanding”... The only way to “conceptualise” (but the Hegelian one is no ordinary concept) change and to conceive of the plurality as concrete rather than abstract, that is, as a pure sum of the unit, is the Hegelian
synthesis... Hegelian logic is not a solution of the paradox but a dismissal of the logical coordinates that generate it."
But Hegelian dialectic remains highly controversial, and perhaps the Aristotelian solution is still the majority position today. We did modify the specifics, see Why is Aristotle's objection not considered a resolution to Zeno's paradox?, after Cantor we no longer reject the actual infinity, but we still believe that there is more than one way to be. Indeed, independent reasons to deny panlogicism, in both Parmenidian and Hegelian forms, were pointed out by existentialists, among others. But there is still one puzzle left: even without the panlogicism mathematical treatment of motion seems to contradict C1. Indeed, mathematics is not just experiencable, and if it is not thinkable then what is? Papa-Grimaldi even charges Aristotle with being unable to accept the logical consequences of "many is not one":
"Can we conceive of anything that does not occupy a space equal to itself at any moment? Hardly (in an ordinary logic, at least). This is the real premise... most of his interlocutors would easily agree on this premise though being unable to accept its logical consequences. Aristotle was one of them.")
The answer seems to be that there is not only more than one way to be, but also more than one way to be thinkable. The transition from one to many or vice versa may well be opaque to the conceptualizing thought Parmenides had in mind (and perhaps also Hegel, despite disagreeing with him), but mathematics shows that even that which is not "thinkable" can be manipulable enough. This brings us back to the "Pythagorean pretence", and to Heidegger's contrast between "calculative" and "meditative" thinking in Gelassenheit (translated as Releasement):
"This calculation is the mark of all thinking that plans and investigates. Such thinking remains calculation even if it neither works with numbers
nor uses an adding machine or computer. Calculative thinking computes. It computes ever new, ever more promising and at the same time more economical possibilities. Calculative thinking races from one prospect to the next.
Calculative thinking never stops, never collects itself. Calculative thinking is not meditative thinking, not thinking which contemplates the meaning which reigns in everything that is. There are, then, two kinds of thinking, each justified and needed in its own way: calculative thinking and meditative thinking."