I am a U.S. citizen and my parents are migrating to the U.S from abroad. Can they move here immediately after their green card is approved?

No, they'll need to wait for the actual green card to be issued and received before entering the U.S. as permanent residents

Once their green card is approved, they can apply for a temporary visa or advance parole to enter the U.S. before receiving their physical green card

It needs to be issued first, not just approved