It would depend on which ‘Persian Empire’ you refer. The Sassanid dynasty was particularly intolerant to everything aside from the royal Zoroastrianism and Judaism. However, the Arsacids were certainly somewhat tolerant, though much of this is simply that their royal religious adherence was different.
The Arsacids in particular, were practitioners of Iranian polytheism and Akkadian polytheism. Their coinage and patronage is associated most greatly to the worship of these gods.
The Great Gods of ancient Mesopotamia reached a revival of sorts under Arsacid rulers. The decentralized nature of the new empire and its explicit love for the Great Gods, continued the general incline of their prominence from the Seleucid period onward.
The Great God Ishtar remained beloved in Arbela, her home city where the Arsacids improved her temples. Nurgle, the Great God of Fire, looting, flaying, etc, reached a renaissance under Arsacid rule. In fact, Nurgle had been one of the least affirmed Great Gods from 603-300 BCE, his temple became dilapidated under Achaemenid rule. Arsacid rulers seem to have totally rebuilt the structure and expanded the cult city of Cutha into one of the major cities in the empire.
Other gods remained prominent in the region and or were associated to another Iranian deity. Naboo remained a common deity, especially in Borsippa, Sippar, and so forth. Whilst Sîn remained master and lord of Harran and the city of Ur. Nippur’s Duranki temple remained splendid, as Arsacid rulers claimed to repair the greatest temple in the world.
The only gods whom the Arsacid neglected, were Assur, Marduk, Ninurta, Dagon and Hadon... Why is that we may ask? The most sure option, is that these gods though universal Great Gods, were associated to a certain non-Arsacid power structure. They also due to that close association to Assyrian and Karduniashi imperialism, made them difficult to become syncretic with Iranian gods and religions. We have no record or evidence of Arsacid repairs to these temples of these gods, like the Seleucids, they may have even repressed these deities.
This matters for the understanding of Arsacid imperialism and religious practice. Judaism was already common to the Arsacids in Babylonia. As such, the peculiarities will not matter much to the Arsacids. Rather, we would see the Arsacids ruling Judah as a sort of tributary vassal stare as the Arsacids ruled almost the entirety of their empire.
That being, a central royal family who commanded a collection of fiefs and free cities and a conglomerate of retainers. A second, being the noble estates, comprising the many Dahae notables who conquered Iran with the Arsacids. These nobles are called Shahs and are kings of their lands. The final group of states are Arsacid vassals and tributaries. These are varied kingdoms that are not nobles but are otherwise subjects if the Arsacid but not under direct jurisdiction of the Arsacid king. These include:
All of Persia, which was a haphazard collection of priest kings,,
Elam, ruled by a set of mysterious Greco-Elamite kings.
The Atropatene kings, remnants of the old Achaemenid empire.
The Western Semitic-Akkadian stars of Adiabene.
The Kingdoms of Iberia-Colchis.
The Syriac kingdom of Osroene.
The Greco-Akkadian kingdom of Babylon. Used as governors.
In short, there may not be a great level of Jewish regional dissent. Arsacid rulers were quite accustomed to ruling a country that was otherwise independent. Arsacids would only request either tribute or warriors wage raids upon Roman lands.
So, under the Arsacids, Christianity may take different shapes. However, its basis may remain the same. In which case, I would expect the Arsacids to pay it no mind generally, unless the religion begins to cause issues in Mesopotamia. As the Arsacid legitimacy relied upon maintenance of temple structures and divine kingship rule in Iraq; they may take much greater action upon Christians who attempt to weaken the power of temples within Babylonia. Likewise, the Arsacids would not take kindly to a religion that operates without temples and generally accepted modes of discourse.
