I learned in secondary school that the Achaemenid dynasty didn't really care about converting others to their faith, and that's largely why they were so tolerant of their conquered subjects and their different cultural traditions and religions.
I also learned at university that the Byzantine-Sassanian Wars are largely considered the first war primarily over religion, where empires fought not primarily for their king, but to prove they have the superior religion.
So during the Sassanian dynasty, was Achaemenid tolerance of religion and lack of evangelism swept aside? In a war of religion, did the Sassanians convert their populace en mass to Zoroastrianism to better help fight against the Christian Byzantines?
That the great Byzantine-Sasanian war was mainly about religion is not really correct. The entire 400-year history of the Sasanians is fraught with frontier wars against the Romans. The professed ideologies of both Rome and the Sasanians were essentially incompatible with posing a mutual existential threat.
But religion did feature in it. However, it must be understood that Khusrau II already ruled over a large, thriving Christian minority in Mesopotamia and Armenia. His pillaging of the true cross may just as well have been intended to produce a trophy for his Christian subjects as humiliating the Romans. It is moreover likely that the large number of Christians Khusrau was subjugating were a problem to the Zoroastrian nobility, whose withdrawal of support is what really caused the failure of Khusrau's campaign.
However the Sasanians didn't try to convert everyone; the main evidence of attempts at conversion is in Armenia, which converted to Christianity from some form of Zoroastrianism while under nominal Sasanian dominion. They did however strongly oppose conversion by Zoroastrians to christianity, indeed imposing the death penalty for it.
As argued by Payne in A State of Mixture, the main concern for the King of Kings was ensuring the subjugation of his Christian population, and it not threatening the balance of power between the House of Sasan and the Arsacid Great Houses.