During the Islamic invasions, all three fell under Arab control. Other than changing their writing system and religion, Persia to this day retains a different ethnic identity and language than the rest of the "Arab world".
The conquering Arabs initially intended for Islam to be the religion of the military conquers, but for the conquered to retain their original religions. For this reason, people were not forced to convert even when militarily conquered. Instead, they had to pay that jizya, or poll tax and lived as dhimmīs, or a protected class. I’m going to quote below something I wrote up on the jizya before.
I was searching for another post when I came across the above comment that I believe is providing some insight for your question. It's the third paragraph in his To the Levant and Beyond section of his comments
That is from /u/frogbrooks comments here; https://www.reddit.com/r/AskHistorians/comments/5mpi1g/to_what_extent_was_the_spread_of_islam_bloodless/dc5gsp7/