How was the Persian religion used as a unifying factor throughout the Persian Empire?
This is a great question. The Persians had their own religion. It was called Zoroastrianism. It was based on the teachings of a man called Zoroaster, and it probably started around the sixth century BC. What is interesting about the Persians is that even though they were a imperial power and had an empire of conquest, they usually allowed conquered people to worship their own native gods. This can be seen as a form of religious tolerance.
A good example of this will be when King Cyrus allowed the Jews to go back to their own land and even rebuild their temple. This caused the Jews to be thankful to Cyrus and even see him as a savior of sorts.
The flexibility and assimilating/accommodating quality of Persian religion allowed the Persians to bring unity in the lands of Persia. Conquered people would be much more favorably inclined with this religious freedom.