Persian cultural influence will be minimal. While undeveloped land tends to inherit the cultural influence of the nearby dominant power, existing developed areas tend to retain their own culture. Mesopotamia still retained its indigenous culture. Egypt remained Egyptian. So conquered Greece will still remain Greek in culture even if there s a greater Persian influence.
Only if Persia retains control for a very, very long time will Greek culture become influenced by Persia. This is a time when cultural transfusion was very limited. Compare with the Hellenestic age. Greek culture spread far and wide, but it was very superficial. Seleucid Persia, Ptolemaic Egypt, and Bactria didn't become Greek culturally even if the political elite were ethnically and culturally Greek for a while.
A Persian satrapy over Greece will probably distort Greek political culture if kept long enough, but there are plenty of Greek areas outside Greece that will continue to develop (Sicily, southern Italy, southern Gaul, Crimea). For that matter, Italy also has a very strong culture at this point, as does Carthage.
Persia did not found colonies like Rome did, nor did it have the same institutional nature of laws, or the ability to assimilate existing cultures to Roman identity. Non-Persians could not become Persians in the same sense that Gauls, Greeks, Numidians, and Jews could later become Roman citizens. Nor would Persians want to give non-Persians such an identity because the aristocratic elites wanted to preserve their role in the Empire. Instead, Persia was much more content to let the lands it conquered continue to operate as before as long as taxes and tribute kept flowing.
Wholeheartedly agree.
But...I wonder what would be the impact of a Persian Greece to the development of Roman Republic/Empire...?