They split as in "went their separate ways".
Dzogchen is based on Indian Antecedents which then were further developed in Tibet in complete absence of any influence from Zoroastrianism, which by the time period you are talking about. We do not have any record in Tibetan, Chinese or any other language of a major contact between Tibetans and Zoroastrians. Zoroastrians strictly do not proselytize. We do have some evidence of contact between Tibetans and Manichaeans in Khotan however.
To clarify, DC and Z share unique ideas. This is from some where. If not from Z itself, then from what came before and it was so old it was just like the basic world view, or there was some cross pollination with Persian ascetics who must have crossed paths with Shang Shung meditators. Somehow these ideas stayed in an area and guys messed with them. But guys in Mexico and Africa, not so much.
If you go to a local Hindu temple and do an astrological puja, the Brahmin priest may give you rainbow colored threads representing the 5 elements.
One blue thread, one red thread, one yellow thread, one green thread, one white thread twisted together.
This is standard Indian Dharma.