The Parthians (and the Sassanids) defenitely did charge with their cataphract. I'm guessing the Seleucids did too, since the whole idea of "shock cavalry" was a Macedonian innovation- but the Seleucids didn't rely on them to a significant extent.

Of the Romans, as far as I know ataphracts were only employed by the eastern Roman empire (later called Byzantine empire) but they were employed like heavy horse archers, much like ancient Persian cavalry. They relied on (german) mercenaries to charge home.