Does an extension cord need its own GFCI if it's plugged into a GFCI-protected outlet?

No. If the extension cord is plugged into a GFCI-protected outlet, the cord itself does not need an additional GFCI. The only thing to watch is the cord’s condition and rating — a frayed or indoor-only cord used outdoors is still a bad idea, even with GFCI protection. GFCI prevents shock but doesn’t stop a fire from a damaged cord.