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Parodied is making fun of something that was serious in it's original work. It's a direct reference to another work.
Played for Laughs is just anything that's done for comedic effect.
Reality is that, which when you stop believing in it, doesn't go away. -Philip K. DickFurther to the above, Played for Laughs usually (though not always) applies to tropes that, when played straight, are more serious in nature. The opposite is the case for Played for Drama. Parodied Trope however does not make that distinction, and can apply regardless.
My name is Addy. Please call me that instead of my username.Parodied Trope is mocking the trope itself. Played for Laughs is imagining if a trope led to comedic consequences.
I'm on the internet. My arguments are invalid.

Is there a difference between "Played for Laughs" and "parodied"? They seem like they ought to be interchangeable, but I'd like to make sure.