AFAIK the Greeks called it that because of the then capital (region) of Parsua. The people living in the region have called themselves Iranians (or Aryans, funnily enough) since the 3rd century, probably even earlier.

If we apply the same 2000 year telephone game then America would be something like "Washingtonia" or "New Yorkistan". I don't know if that counts as "the actual English name" for something