Against Egyptians with Parthia.. The only thing you have that can effectively kill chariots in melee is elephants.. Too bad I guess. You can pin them with Desert Cavalry and then charge them with Catas, but that path is costly..
What about infantry? A large mass of semi-upgraded Eastern Infantry can take down a few chariot units. Chariots are vulnerable to even weak infantry. Especially when they are spearmen and recieve a combat bonus, but of course depending on what difficulty you play, your Eastern Infantry can rout quite easily, because chariots already cause a devaluation in morale to foot units.