Mani was supposedly raised in an obscure Jewish-Christian sect in Persian Babylon, which he later distanced himself from to found a new religious movement which blended Gnostic Christianity with Zoroastrianism and, ostensibly, Buddhism.
Given that he spent much of his life in Persia and Mesopotamia, and was patronised by Shapur, it makes sense that Mani would have been intimately familiar with Zoroastrianism. Gnostic Christianity, which was also dualistic, would have been easily paired with Zoroastrian philosophy, but Buddhism is a rather different tradition entirely, and one which, while present in the Sassanid Empire, would likely have been far more distant and alien to Mani than Christianity or Zoroastrianism.
Nonetheless, Mani seems to have considered Buddha, along with Zarathustra and Jesus, part of a succession of prophets that each revealed part of the truth. Do we know how how Mani learned about Buddhism, and the extent to which he was familiar with it? Do we know enough about the teachings and worldview of Manichaeism to pinpoint ideas and practices in the Manichaean religion that were Buddhist influences? Indeed, other than the obvious inclusion of Zarathustra in its line of prophets, what were the Zoroastrian influences?
This is a good question, and I think the first one is to sort out what we are talking about when we say "Buddhism". The territory of Shapur's empire stretched to the Indus river, and included the territory annexed from the Kushan kingdom. Now, the Kushans seem to have practiced a syncretic form of Buddhism that mixed influences from Hellenic, Iranian, Indian and even Mesopotamian religion. Buddhism, which has a long tradition of syncretism, as practiced generally differs a lot from the more philosophical "textbook Buddhism" most people are familiar with.
Mani is said to have spent time with learned men (I believe the term used is "Brahmins") in the far East of the Empire, and so he certainly had access to Buddhist thought one way or another. When it comes to what Manichaeism seems to have borrowed from it, what is usually pointed to are confluences with Gnostic Christianity: the flesh is corrupt, worldly pleasure is to be eschewed in favour of asceticism, and from this there is a general misogynistic streak portraying women as wicked temptresses.
These can also be viewed as points of contrast with Zoroastrianism - there, the flesh is corruptible but not inherently corrupt, pleasure is viewed as a good, and while there is misogyny it is not as deeply rooted and more of a patronizing and paternalistic kind, viewing women as weak and susceptible to corruption, rather than as corrupters of men. Where Manichaeism does borrow clearly from Zoroastrianism, this is more in cosmology and also not always easy to divide from the "Gnostic" influences - good and evil as polar opposites, a world divided into three stages (beginning, middle, end), etc.
It's not really widely agreed upon how much Mani was really drawing in Zoroastrian ideas, versus casting his notions in Zoroastrian terms, because a lot of his usage of Zoroastrian terminology seems pretty superficial. He uses "Az", or "the Serpent", as the name for the primal temptress, and Gehmurd (Zoroastrian Gayomard/Gava Meretan) as the name of the first husband, and these do not really match up with their usage in Zoroastrianism for the world-destroying serpent of the apocalypse Azdahag, and the primal, perfect man (the first husband is an entirely different figure in Zoroastrian cosmogony).
So, I think the overall takeaway is that while Mani was obviously influenced by his patronage from a Persian king, he seems to have mainly drawn on influences to the extent where they helped shore up ideas he was already familiar with from early Christian sects. Although, influences back and forth between early Christianity, Hellenic mysticism, and Persian religion is pretty muddled and confusing, so it's not always really possible to sort out what comes from where.