Ah, yes. It makes sense that you can't play the Overloard as a Pirate Reaction.. The Overlord can play an action, not a reaction.
More accurately, there is no rules distinction between what part of Pirate's text is the "Action" and what is the "Reaction". The entire card is always both an Action and a Reaction (and also a Duration). When you are playing an Overlord, such as during your Action phase by spending an action as per the rules of the game, you
can use it to play a Pirate from the supply — Pirate is an Action and is not a Command, and it being other types in addition to Action doesn't stop that from working. When you do that, the Overlord plays a Reaction, because it plays a Pirate, and Pirate is always a Reaction.
The reason why what you're trying to do doesn't work is that Pirate's below-the-line ability lets you play it from your hand; it does not let you play cards other than itself from your hand, the Overlord that you do have in your hand is an Overlord, not a Pirate, and Overlord does not have any special abilities that let you play it at unusual times like when someone is gaining a Treasure. This is always true regardless of whose turn it is.