There was a question several months ago on whether there was any ancient civilization that didn't have slaves, period, and the answer was "Persia, maybe." Unfortunately, I can't find the thread, but here's a related answer explaining that slavery might have been banned in Persia for religious reasons - but we don't know for sure:

Zoroastrianism does indeed ban slavery, but Zoroastrianism did not exist as an organised religion in c. 480 BC to the best of our knowledge; contemporary documents clearly indicate the continued use and sale of slaves in Mesopotamia and the Greek city states in Anatolia during their period of rule, most particularly in Mesopotamia where we have economic documents.

However, it is not implausible that there was a ban on slaves within Persia itself, and other Iranian speaking regions. The problem is that archaeologically it is very difficult to evidence slaves and slavery. Literary evidence is generally how we find out that a culture practised slavery; Linear B tablets are why we know the Mycenaean Greeks kept slaves, for instance. And we possess a much smaller corpus of Old Persian texts than Akkadian, for instance.