Iran’s position at the intersection of Mesopotamian, Arabian, Central Asian, and Indian cultural currents (among others) has influenced its astrological systems. As such, I don’t know that it’s quite accurate to speak of a Persian zodiac. Astrologers in the Iranian cultural sphere have made extensive use of both the Babylonian (“Western”) zodiac of twelve month-signs and the Sino-Turkic (“Eastern”) twelve-year animal cycle--which, strictly speaking, is not a zodiac. The former is almost certainly the older system in an Iranian context, but the latter has a long history of use as well.

In an Encyclopædia Iranica article on “Zodiac,” Antonio Panaino writes that, while no surviving Avestan or Old Persian sources relate directly to astrology, there is every reason to believe that Babylonian astrology was both known and practiced in ancient Iran. This system was interrelated with and influenced by Greek and Indian astrology, which both use essentially the same twelve-sign system, pegged to the constellations along the “path” of the sun. By the time astrological writing begins to appear in (Middle) Persian, during the late Sāsāniān and early Islamic periods, we find the twelve-month system fully incorporated into Zoroastrian cosmology. Classical Islamic astrology also employs this system, drawing extensively on Greek models in particular, and classical encyclopedia works like the ‘Ajā’ib (“natural wonders”) corpus discuss the heavens and calendar in these terms. Iranian months, in fact, correspond directly to zodiac signs (so for instance, the month of Farvardin usually begins on March 21 and ends on April 20, corresponding to Aries/borj-e hamal/barreh).

The “Eastern” twelve-year system was known amongst steppe peoples, such as the Uighurs and Khotanese, by the eighth century CE at latest. As Turkic groups regularly settled in, traded with, raided, and conquered portions of Eastern Iran (and Persianate Central Asia) throughout the late antique and medieval periods and beyond, this system would have been present at least sporadically in these regions. However, Charles Melville dates official adoption of what he terms the “Chinese-Uighur Animal Calendar” in Iran to reign of Hulagu (1256-1265), first of the Mongol Ilkhanid dynasty (that article can be found here). Although initially used only to record events related to the Mongol rulers, it came to be widespread among the population. (It also probably helped that most rulers of Iran from this period onward were partially, if not entirely, of Turko-Mongol descent.) Melville quotes a seventeenth-century Iranian historian who claimed that by that point, most people only understood reference to “Turki” years. It was only in 1925 that the official usage switched to the “Solar Hejri” calendar in use today in Iran, though apparently some areas of Afghanistan continue to use the twelve-year animal cycle (see here.)

...And I know you specifically didn’t ask about the whale-dragon switch, but I’m going to answer that anyway! I touched on this subject in this previous answer of mine, about dragons. Essentially, it seems that the creature referred to in Chinese as lóng and in Mongol as lū was translated into Persian as nahang, a word referring to dangerous water beasts both real and legendary (crocodiles, hippos, sharks, sea serpents, etc.) Only in the the 20th century did nahang shift to referring specifically to cetaceans, though it retains the more generic and monstrous sense in classically styled poetry.

Please let me know if I can offer any clarifications or follow-ups!