I want to know how to say "Eleatic arguments" as well as how to say "Eleatistic arguments". Right now, all I can come up with for the former is "argumenta Eleatica", and I have no clue about the latter. Thanks in advance for your help!
"Eleatica" indeed seems to be the correct way to refer to the "Eleatic school," founded by Parmenides.
This book includes on pg. 62 a section titled, De Secta Eleatica.
Although I can find no written examples in medieval or modern Latin of eleatistica (by which I presume you mean "Eleatic-imitating"), I think a convincing analogy can be made with the more common terms "Hellenic" (hellenica) and "Hellenistic" (hellenistica). The same transformation yields eleatistica.
You thus have: