There are basically three types of parody, and that guides what you do with a parodical story.
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You are closely parodying a well-known story or beloved fandom.
This is where you try to write a story about “Star Battles” or “Baconlet”, and most of the humor comes from adding twists to the known structure of the story. This easily falls flat. People have to have read or experienced what you are talking about to get the joke, have to appreciate your re-telling of the story, and have to not get bored with your single-minded subject matter. It’s possible to do this well, but it helps to avoid cliches and include as much variety about the setting/fandom/genre as possible to break up jokes based on the single story. -
You are parodying a well-known genre/fandom and its tropes.
This is where you write a self-aware horror story, dating sim, or game about geeks that makes fun of itself. While easier to make accessible and funny, it’s prone to in-jokes, obscure references, and lacking a coherent plot apart from its sarcastic goal. For a great example of how to do this right, go play the demo of Don’t Wake Me Up, which is full of nerd references and more generally makes fun of video games. -
You are making fun of something that exists in the real world.
This is more accurately called “satire”. It’s the least “niche” of these types of parody, accessible to the most readers, and is more like a comedy sketch. It takes a social norm or a commonly held idea and turns it on its head. Most writers can get away with this kind of comedy, so it isn’t as rare. To pull it off, you must avoid copying people (using cliches), say something reasonably insightful, and being excessively cruel or sarcastic will lose you some of your audience.
Hope that helps.