This kind of depends on what you mean by "bronze age Iran." The geographic area of modern Iran didn't host the loose ethnic/linguistic/cultural group of "Iranians" until the very end of the Bronze Age. Even then, it took a couple of centuries before identifiably Iranian groups reached the western side of the Iranian Plateau and the Zagros Mountains and entered into the written records of the Neo-Assyrian Empire.

We know relatively little about any of the cultures that inhabited geographic Iran during the Bronze Age. The eastern half of the region is functionally unknown outside of material culture uncovered by archaeology. We can certainly speculate about how those material remains indicate the existence of a class system or religious hierarchy but cannot say anything definitive from that evidence alone.

The western side of the region is only marginally better, known primarily through the local culture's interactions with their Mesopotamian neighbors. Notable examples include Lullubi (northern Zagros), Marhasi (western-central plateau), the Gutians (southern Zagros), and the Kassites (also southern Zagros). What little is known about these is usually in reference to their kings or chieftains. A few Lullubian monuments with inscriptions have been identified, where their kings invoke gods and worship, but little is known about their actual social or religious structures. Kassites certainly integrated into the Babylonian religious hierarchy after conquering southern Mesopotamia in the 16th Century BCE, but not much is known about their homeland.

Likewise, both Kassite and Marhasi names have been speculatively linked to the Hurrians of northern Mesopotamia and Syria, which certainly had a class structure and priesthood in those regions, but that does not necessarily say anything about their potential connections or roots in the Iranian region. On a related note, the ruling class of the Mitanni Kingdom that dominated the Hurrians in the Middle Bronze Age had some level of cultural and linguistic roots in the Indo-Iranian cultures and likely began migrating from Central Asia at the same time their linguistic and religious relatives migrated into the Indian subcontinent. They too had a religious hierarchy by the time they reached Syria, presumably a similar society to the early Indo-Aryans reflected in the Vedas, and must have passed through Iran at some point, but we don't know what that period in their history looked like either. The presence of kings and other political leaders in these cultures clearly indicates some sort of class hierarchy, but how formal that was outside the immediate royal sphere is, predictably, unknown.

The Elamites of southern Iran, largely corresponding to modern Khuzestan and Fars, are the best documented Bronze Age culture in the area. Elamite written records, though few compared to their Mesopotamian contemporaries, at least survive to provide a glimpse of their society. In part because of frequent contact and occupation of, and by, their Mesopotamian neighbors, their social and political hierarchy was largely similar to that of southern Mesopotamia. There was a hierarchy of royals and nobles both on a city state level and as part of larger kingdoms as they rose and fell. For about 400 years between 1900-1400 BCE, this hierarchy was even disrupted at the highest level when the Sukkalmahs, a title meaning something like "Grand Viceroy" or "Prime Minister" actually displaced the Elamite kings as the primary political leaders. Interestingly, the Sukkalmah period also includes the greatest moment of Elamite expansion under Sukkalmah Siwe-Palar-Huppak.

Also like Mesopotamia, they had formal temples and priesthoods, including Chogha Zambil, the largest known ziggurat. Priests officiating ceremonies are a routine feature of Elamite artwork. Their priests had a variety of titles indicating different roles within the priesthood. Some of these titles do not have a clear meaning in the surviving records, like Pirramada. Others are more obvious. A Liriria was an officiant who oversaw specific types of ceremonies. For example, a common type of religious festival known as a Lan was directed by a Lan-Liriria. The highest ranking priests were called Shatin and tend to be associated with overseeing temples or rites of the most important gods.