Persia was never part of a caliphate.
Define "part of a caliphate."
In terms of being under direct political and economic hegemony of a caliphate, this statement is demonstrably false (re: the Rashidun conquest of the Iranian Plateau). In terms of pleading fealty as a vassal to a later caliphate like the Umayyads, Fatimids, and Abbasids, again, this is also false (re: Khwarezm being a nominal vassal of the Abbasids prior to the Chinggisid conquests of Iran and Iraq, followed by the siege of Baghdad and the execution of the final Abbasid Caliph).
It's only in early modern history where what we think of as "Iran" or "Persia" vehemently objected to being "part" of a "caliphate," and that's because of the Safavids forcibly converting the Iranian Plateau to Twelver Shia Islam as a direct objection to Ottoman hegemony in the Levant. Later Iranian dynasties like the Qajars, the Pahlavis, and even modern post-1979 theocracy Iran upheld this imperative against Ottoman hegemony and the hegemony of later successors like Turkey and the KSA.
Isn't Iran one of those countries that only exists because Christians were too gracious after WW2?
False on so many levels.
a) Iran under the Sassanians in the 400s accepted Nestorian refugees who fled Rome after the Council of Ephesus which denounced Nestorius as a heretic. Edessa became a hub of Syriac-speaking Christian missionaries who evangelised the entire Silk Road, going as far east as Tang China. Those same Syriac missionaries eventually made their way into the Indian Subcontinent to begin their evangelism. One group went north and stayed along the Indo-Gangetic Plain, the other group travelled southward to the Malabar Coast.
b) Even under Muslim rule (re: post-Rashidun conquest), Christians were
nominally (though not always practically) considered part of the Ahl al-Kitab (Peoples of the Book). Syriac Christians were still able to continue their evangelism for
centuries regardless of
who was nominally declared caliph in Damascus, Baghdad, Cairo, etc. Syriac Christian missionaries in Central Asia became extinct well into the 15th century.
c) Governance of the various Iranian empires across history was decentralised to a certain degree. You had imperial governors (satraps) appointed by the emperor, and given responsibility for a province (satrapy). This was the state of affairs for
all Iranian empires, not just the big three of the Achaemenids, Parthians, and Sasanians. It's also a huge influence of regional culture to this day. Those historical maps of past Iranian empires look impressive until you realise that it's all basically a network of imperial governorates held together by the authority of a powerful emperor. Even if the emperor wanted to kill all Christians, he realistically couldn't make that happen if the governor either said "no" or just lied and said he would.
d) Iran was a predominantly Muslim country for centuries that secularised hardcore during the Interwar period thanks to the British ousting the Qajars and installing Reza Shah Pahlavi on the throne in the 1920s. Iran was nominally an allied nation throughout all of WW2 because of British and American interests in the country. The instability surrounding the modern nation-state of Iran happened during the
Cold War. Long story short: repressive foreign puppet regime in the Pahlavi dynasty got ousted, new democratically elected leader shows up and makes a big upset, big private foreign interests hate this, they go to America, they have the CIA oust Mossadegh and reinstall the Shah.
e) Imam Khomeini was not the architect of the 1979 Iranian Revolution; he just hijacked it after France flew him back. You wanna blame someone for the sordid state of affairs for non-Muslims in contemporary Iran? Blame the French for doing huge photo ops with Imam Khomeini before flying him back to Iran despite
countless Shia clerics opposing it because Khomeini was way too damn radical.