The story about the Magi visiting Jesus comes from Matthew 2: 1-12, no other Gospel makes mention of this event. It doesn't mention how many Magi there were, where exactly they came from, or when they arrived. It's not widely accepted to be historically accurate either. That makes interpretation of the story a little tricky, and very much dependent on assumptions.

If you were to start from the assumption that it did happen, the question becomes "where did these magi come from, and why did they show up?", and from there the leap to Parthians isn't that great.

Personally, I tend to start from the assumption that it didn't actually happen, which means the relevant question becomes "why did Matthew write this story?" And then, I think, including Parthians makes much less sense.

All four Gospels play games with themes of kingship, and it's fairly clear this story fits in with that: people coming to pay homage to a new king, with what are essentially royal gifts. Matthew even has them arrive at Herod's palace to press home the point. The big question to me, is not so much where they were from, but why they were magi at all. All other mentions of magi are not exactly positive. They're magicians, fortune-tellers, people practicing forbidden arts. Anyway, I kind of want to leave it at that question, because that's where I move from knowing a bit to full-blown speculation and things I read on Wikipedia...