Why is it that Zoroastrians influenced Judaism? Wouldn't it be the other way around?
It depends on who's writing the history books. I had a Old Testament professor in college, who taught that the Egyptian pharoah Akhenaton "invented" monotheism. He claimed the Israelites first "discovered" monotheism from the Egyptians during their captivity, and then retroactively took credit for it when writing the Pentateuch.
No. Most theological history puts Zoroastrianism as a sort of pre-cursor to Judaism from which Judaism acquired many of its modern characteristics, arguably the conceptual grandfather of the major Semitic religions (Judaism, Christianity, Islam). The arguments are not whether Zoroastrianism influenced Judaism, but to the extent that Judaism is directly descended from Zoroastrianism in the same way that Christianity is descended from Judaism. It is well-established that many parts of Judaism were acquired from the Persians, but the exact relationship in pre-history is uncertain.
Zoroastrianism is an extremely old religion, older than recognizable modern Judaism for sure, though Islam pretty thoroughly eradicated it. Most major modern religions are at least partially descended from it.