Umayyad Caliphate vs Persian Empire : who would have won more?

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By "Persian Empire" do you mean the Achaemenid Empire (558–330 BC) or the Sassanid Empire (224–651 AD)?

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This.

And won more what?

If Sassanid, then the answer should be obvious, considering it was the burgeoning power of the Umayyad Caliphate that dominated both the Sassanid and Byzantine Empires. Though even that is truly fortune. The Byzantines were considerably weakened after losing 1/3 of their population to plague in the mid-6th century, followed immediately by a 2 decade long brawl with the Sassanids, so you can diminish the Umayyads by saying they just cleaned up the tired and beaten armies of both.

If the Achaemenids, then I'd have to say the Achaemenids at the height of their power would dominate. The considerable manpower of the satrapy system provided for some of the largest armies of the ancient world with some of the best military infrastructures. The style of war waged by the Umayyads would not counter well the Persian fronts, unlike the Macedonian system seen in the 320s.

ETA: I'd love to ponder a Justinian-era Byzantine army led by Belisarius against the armies of the caliph.