Achaimenid name for themselves?

[Zenodoros]

But that's using the Greek/Latin names rather than Persian names. Is that how he actually wrote it?

They called themselves Persians (Old Persian Parsa) and Aryans (Old Persian Ariya).

I thought Parsis was just one area Cyrus gained control over, and one that the Greeks learned about early-ish on, so they attributed that name and those derived from it to all Achaimenians.

Would Haxmaneshya be the Persian equivilent to "Achaimenian"?

Thanks for your help.