As most things are, this is somewhat correct and otherwise false. It is true that Christianity in particular began to spread far across the Persian Empire. This can be interpreted to mean that Zoroastrianism was "dying" only from the perspective of ignoring key historical information. Here are some of those historical informata.

(1) The other large superpower at the time, the Roman Empire (or Byzantium if you're inclined to follow standards of Medieval European schoalrship in the 21st century) was busy in a back-and-forth debacle of declaring half the Empire a heresy, and the other orthodox, and then back and forth. Here's a list of the some contributing events:

• First Council of Nicaea (325) repudiated Arianism (and exiled them! We'll see this is going to be a common pattern).

• First Council of Constantinople (381) repudiated/exiled Arianism and Macedonianism, repudiated/exiled Nestorianism (for the first time, stay posted for round 2 later) , proclaimed the Virgin Mary as the Theotokos ("Birth-giver to God", "God-bearer", "Mother of God"), and thus repudiated Pelagianism (and exiled them , and reaffirmed the Nicene Creed (and exiled whoever didn't!)

• Second Council of Ephesus (449) declared Eutyches orthodox and attacked his opponents. (hence exiling them) Though originally convened as an ecumenical co Council of Chalcedon (451) repudiated/exiled the Eutychian doctrine of monophysitism

• Second Council of Constantinople (553) repudiated and exiled the Nestorians. The Nestorians and their affliated predecessors the Arians were MAJOR movements, with PROMINENT leaders and WIDESPREAD support. TO exile them and make them unwelcome is to exile lots of people.

• Third Council of Constantinople (680–681) – repudiated/exiled Monothelitists and Monoenergites

• Second Council of Nicaea (787) – exiled iconoclasts. This is really fun, since the anti-iconoclasts would soon exile the iconoclasts, and vice-versa for a third round. If you never believed in Hegelian synthesis, the Roman iconoclast fiasco is your best bet.

Anyway...

These exiled/repudiations might have involved hundreds of thousands (or thousands) of people, and the only civilized place in the world to go at the time was probably Persia, which leads me to the following:

(2) The Persian Empire was notoriously tolerant. The basic requirement of any religious or ethnic minority was to keep their members in line, pay taxes, and avoid dissent. And in 410 AD, during the rule of Yazgard I (399-420), Christians were recognized as a millet, or separate religious community, and were protected as such within the legal framework of the Sassanid Empire. So whereas they might be killed, robbed, exiled, or otherwise slandered (the meaning of "repudiation") in the Roman Empire, as long as Christians respect the rule of law, they could live, preach, and travel as they wished in Persia. This is also how the Central Asian Turkish tribes like the Naimans became Nestorian Christian, or how some of the earliest Christian communities in China and India started.

Conclusion There was conversion and Christian immigration into Persia "even before the Arab conquest" but I fail to see how or why this means Zoroastrianism was in the process of declining. If anything, the growth of Christianity shows the tremendous strength of Zoroastrianism since the Persian government did not feel threatened enough to institute oppressive policies against (most) religious minorities. But this is an old line in scholarship, and you can find it in Phillip Hitti's History of the Arabs (1955 edition) and others. It's an old and tried hypothesis, and one no longer supported.

Readings and Sources:

"The Spread of Christianity" (pp. 36-43) From:Cultures in Motion Peter N. Stearns Yale University Press (December 2001)