So for example, Spaceballs the movie is a gigantic parody of the Star Wars trilogy. It seeps from every pore of this film. The Princess has buns over her ears (a la Princess Leia) which turn out to be headphones. The "Han Solo" character has a half-man, half-dog fuzzy co-pilot, making fun of Chewbacca the Wookie. Instead of the Force, they have the Schwartz.
So it clearly is "based on" Star Wars, but it is done to make fun of it and be funny. It is like Saturday Night Live doing a skit to make fun of a Richard Simmons exercise video. It is in no way meant to steal (plagiarize) the content or have any confusion about being written by the real author. It's meant to be funny.
In comparison, with a homage you want the reader to have appreciative thoughts. So if there is a classic scene in a great romance novel where the couple kisses while the "world spins around them", a modern romance movie might have this same scene laid out in order to evoke that same feeling.
Parody is one of those thin line areas with copyright law. With a homage, it is usually just one scene that is a homage, done in a specific way. You don't do an entire movie with the exact plot and dialogue and layout of Die Hard and call it a homage. If you make a new version of Psycho with the exact scenes and dialogue (which was done), that needs to be negotiated for rights.
But a parody - for example "Like a Surgeon" song which parodies "Like a Virgin" - you *technically* don't need permission to do that. However, Weird Al does get permission for every song he runs, as a professional courtesy and also for legal reasons. If someone really did want to sue you for theft, they probably could. Like in all things, it would depend on the situation.
For example, a laid back sci-fi writer might enjoy that you did a parody of their book ("Doon" was a parody of "Dune"). But if someone wrote a very strongly Democratic book and then someone else wrote a parody with the sole purpose of bashing Democrats and making them look like dunces, lawyers might be called in.
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