Can I play spells with flow keywords like Perfect Execution during my opponent's turn?

No — Flow doesn't change when you can play a spell, only where from (your trash instead of your hand).

Perfect Execution has no Action or Reaction keyword, only Flow, so it's a "base speed" spell (Rule 155): it can only be played during an Open State on your turn outside a Showdown. Rule 829.1.b.2 is explicit that playing for the Flow cost doesn't alter timing permissions.

So on your opponent's turn you can't play it — not even during a Showdown, since that requires the Action keyword. For contrast, a spell like Defy (which has Reaction) can be played from your trash via Flow on your opponent's turn, but only in response to a Chain item or during a Showdown.