Egyptian Coptic Christians unironically welcomed the Arab Caliphate since it "freed" them from the "tyranny" of their Greek Orthodox Easter Roman Empire overlords.
Baaki ka Chronology aap ko to pata hai.
When Caliphate boys came, Sassanid nobility was fighting each other over who gets the kursi, fire-worshipping be damned.
Infighting among kuffar tribes is the biggest ally of Islam
Its not just fighting for the Kursi that doomed the Iranians, what doomed the Iranians ( and ultimately the Greeks, aka the Byzantines), was the MOAW - the Mother of All Wars, as it was called by the Byzantine sources- the 605-630 CE ( or thereabouts) war that ruined both the empires, sapped their manpower & economic strength, that saw the Byzantine Empire nearly die ( for a 5-6 year period the House of Sassan had nearly recreated the Persian empire under the house of Sakhamani - Egypt, levant & most of Turkey under Persian control, with the byzantines cut off from their asian assets), then the Persian empire suffer a daring & destructive hit to its heartland ( a path of destruction, carved from basically Georgia, right into the Persian heartland & into Iraq) that ensured Byzantine survival & status quo returning to pre-war (on paper), resulting in severe depletion of manpower & resources for both.
Within 10 years of this war wrapping up, the Arabs descended on them both, lead by one of the greatest generals in history ( Khalid Ibn Walid). One byzantine source i read stated that Heraclius's army at Yarmouk consisted mostly of fresh raw recruited teens, while the arabs fielded a much smaller army but with 'almost every man being 30 or older).
If mohammed had been 20 years older or younger, Islam would've been a fogotten religion, never managing to break out of the arabian peninsula in the first place. but once it did, it spread rapidly in Persia because at the end of the day, Zoroastrianism has the same set of ethics as Abrahamism, as it is an older form of monotheism ( something that greatly influenced Judaism for eg) and shared a lot in common in mechanism.
For eg, Zoroastrians, like these Abrahamics, are the only other religion to state sponsor a religion & go into state sponsored irradication mode for other religions.
persuading 
, in the same way the Masri cope goes like "hehe we went to war just to capture Sinai, and we got it, not destroy Israel and kill all the jews and grab the land"
