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This is a very complex question that cannot be definitively answered, given the long span of time and scarcity of sources. I don't have a good reference work on the Sasanian empire on hand right now, so I'll have to rely on some narrower works (focusing on Zoroastrianism and political ideology plus Payne's "A State of Mixture: Christians, Zoroastrians and Iranian political culture in antiquity"). What religious minorities were present is pretty easy to answer, thanks to an inscription left by the High Priest Kerdir, who pushed for the entrenchment of Zoroastrianism as a state religion in the late 3:rd century. Describing his deeds he says at one point,
And Jews, Sramans (Buddhists), Brahmins ("Hindus"), Nasoreans (Orthodox Christians), (Gnostic) Christians, Maktak (Baptisers), and Zandiks (Manichaeans) in the empire were struck, and idols were destroyed, and dwellings of the dews were scattered, and [non-Zoroastrian] temples and nests was abandoned.
Given the mentioning of numerous Christian groups, we can probably take this list as pretty exhaustive, allowing for great diversity within the Gnostic sects of Mesopotamia. This is undoubtedly the single most debated line in the entirety of Sasanian literature. There are a couple of camps:
The traditional one, which supposes foreign faiths were outlawed. I don't think any sane person still subscribes to it.
The interpretation of Payne, that we should consider these to be three separate events: The "striking" or forcing into submission of accepted but subordinate religions, the destruction of ("pagan") idols, and the abandonment of ("pagan") places of worship.
The interpretation that Kerdir just made this up and nothing actually happened.
The middle of the road camp, where I'd put myself. I don't think Payne's interpretation is viable in light of the fact that Buddhists were associated with idolatry in Zoroastrian scriptures (possible, Buddha was identified as the daeva of idolatry Buiti in the Vendidad, but this is disputed; nevertheless, Buiti's middle persian name was Bw~t, which was conflated with Middle Persian for Buddha or Buddhist idol, Bwt), and would surely have been the first to come to mind when mentioning the destruction of idols. There might have been minority "pagan" or "polytheist" sects the clergy disapproved especially strongly of, sure, but one wonders how widespread they really were. At any rate, I doubt there was a massive coordinated campaign of persecution; I suspect that there were local persecutions and pogroms in places where religious minorities were perceived to have too much influence and that these were encouraged by Kerdir, and that he persuaded the Great King to strengthen the standing of Zoroastrianism and formalize the subordinance of minority religions.
Kerdir's rise to power was also responsible for the martyrdom of the prophet Mani and the expulsion of Manichaeans from the vicinity of the royal court into Central Asia, where the religion would go on to flourish. Before that, under the 30-year reign of the Great King Shapur (who styled himself "King of Kings of Aryans and non-Aryans", a tradition that would go on), there had been considerably more tolerance and Manichaeans had been allowed, even encouraged, within the royal court itself, and the king became a patron of Mani's (though not a convert). Only after Shapur's death and the rapid death of his successor did Kerdir seem to have accumulated enough power to persuade the king into demoting Manichaeism.
Armenians had since Achaemenid times practiced a religion heavily influenced by Zoroastrianism, and were in many ways culturally Iranian (the ruling dynasty was, in fact, a branch of the Arsacid Dynasty of the "Parthian" Empire). This created tensions as Christianity spread through Armenia, which was in a state of intermittent conflict with the Sasanians. Famously, Tiridates III went on to proclaim Christianity the official religion of Armenia in the early 4th century; after Shapur II had subjugated it about sixty years later, he attempted to, unsuccessfully, "de-convert" the Armenian Christians, and it was soon back to being a Roman client state. Only in the mid-5th century was Armenia truly turned into a Sasanian client kingdom ("Marzbanate"), with the Sasanians accepting Christianity as its official religion in the late 5th century in exchange for military help against the Hephtalites (at least according to Armenian tradition). For their part, the Sasanians had pushed the Syriac Christians into establishing a "Church of the East", which went on to declare independence from Rome in the 420's (and later failed to follow the Patriarch's demand to condemn the heresy of Nestorius, leading to the misunderstanding that they were "Nestorians").
Jews also flourished in the empire, primarily in Babylon, where the Babylonian Talmud began to take form. I don't know if their precise legal status is known, but they were permitted to practice their faith, at least, even though relations with Zoroastrian clergy wasn't always friendly. Indeed, much of the Talmudic discussion of purity rites seems to be influenced by interaction with (and opposition to) Zoroastrian practices.
Overall, Zoroastrianism was at this point an ethno-religion, and the debate within clergy appears to have centered on what the spiritual consequences of non-Iranians violating Zoroastrian purity rules was (with some maintaining it was irrelevant, others considering it a necessary evil, others finding it acceptable as long as it was done outside of Fars), so there was very limited interest in converting minorities, though vehement opposition to Zoroastrians converting (including application of the death penalty for this offence). Because of the theocratic and absolutist nature of Sasanian monarchy, policies must have varied a lot with reigns (and, with regard to Christians, with relations to Rome, of course). Some kings, like the famous Khosrow "Anurshivan" ("immortal soul", this epithet is how he is usually referred to in Persian) who reigned for nearly five decades, had a somewhat open-minded view of foreign ideas, maintaining that all wisdom had come from "Iranshahr", regarded as the ancient land of the Zoroastrians. Thus, he is supposed to have reasoned, learned men should be invited so that this knowledge could be re-collected.
Unfortunately, much of our knowledge of the Sasanian empire comes from Christian hagiographies, which are obviously very heavily slanted and cannot be taken at face value. Payne showed, I think convincingly, that these were situated in a distinctly Iranian cultural-religious context, suggesting that relations were not at all as hostile as the hagiographies would suggest but rather that interaction between Christians and Zoroastrians was frequent and generally amicable.
Wrapping up, the Sasanian empire was tolerant enough that minority communities could thrive; hardly the paradise of ecumenism or religious freedom some would suggest, but not the dystopia of Christian sources either.
Sources and further reading:
A State of Mixture by Payne (2015)
Arsacids and Sasanians by Shayegan
Wiley-Blackwell Companion to Zoroastrianism (2015)