First, it’s important to get our bearings re: biblical chronology. The Babylonian Captivity began in 597 BC with the deportation of the king, the royal family, the nobles, the warriors and the artisans. It culminated in 587/586 BC, when Jerusalem, including the Temple, was destroyed. The implicit claim is that the Jews were influenced by the Babylonians, Persians who would’ve been Zoroastrians, then a relatively newly minted religious people.
However, “the vast stretch of Jewish and Christian tradition credits” the Pentateuch (the first five books of the Bible, include Genesis) to Mosaic authorship and being written during the lifetime of the patriarch, according to The Ignatius Catholic Study Bible—Genesis. That would mean “either the 1400s 1200s B.C.” (p. 7).
In contrast, modern scholars put the beginning of the composition at about 900 B.C. and a finish date of about 400 B.C. Yet even “most who adopt this view acknowledge that the stories in Genesis are often much older than their written form, and some would allow that certain parts of its contents may indeed be Mosaic in origin.” Indeed, “Studies of the Book of Genesis within its Near Eastern context tend to confirm both the antiquity and the authenticity of its traditions” (Ibid.).
So, at the outset, we can see that there’s solid evidence for the taking place of major biblical events long before the Babylonian Captivity began, and that would include the Virgin Birth (Isa. 7:14).
In addition it’s true that, according to Zorastrianism, the birth of Zoroaster or Zarathustra was foretold long in advance. And a later tradition has him born of a virgin. And that he overcame all evil opposition. These similarities with Christianity do not mean that Jewish Old Covenant prophecy and Christian New Covenant fulfillment borrowed from Zoroastrian. That would erroneously imply that they were manufactured stories, myths, instead of presenting actual God-ordained realities.
In summary, “the magi [or wise men] from the East who came in search of the newborn King of the Jews were priest practitioners of Zoroastrianism or something like it. So its teachings did dispose them to be open to the Good News of the birth of the Savior God of the Most Pure Virgin,” as my colleague Father Hugh Barbour, OP, says (see Matthew 2). But that doesn’t mean that the Israelites or subsequently their spiritual descendants—Christians—borrowed from Zorastrianism.
For more on Zoroastrianism’s alleged influence, see Dr. Matthew Bunson’s brief overview on the subject, which illustrates that Zoroastrianism is more akin to certain Christian heresies, not authentic Christianity.