Excel Saga doesn't parody anime, it parodies the entire act of watching anime, treating every convention and restraint as permission to break it again. The show burns through jokes at such velocity that half the gags hit before your brain registers what's happening, and the other half hit three episodes later when you realize they were setting up the punchline all along. By episode 13 it literally restarts itself and admits it has no idea where the story is going. This is a show built for people who have watched enough anime to be exhausted by it, and need something that laughs at their exhaustion alongside them.
Excel Saga sets out to parody anime as a whole, and nothing is off limits. Il Palazzo, the mysterious leader of the ACROSS organization, wants to rule the world but decides to begin with the city of Fukuoka. He hands the job to two young officers, Excel and her partner Hyatt, who bungle even the easiest assignments, usually ending in death and widespread destruction. Opposing them are the City Security staff: three fairly ordinary men, a very stern young woman and a few robots. Even the credits are gags, from a spoof of Japanese pop videos at the start to a closing ballad sung by Menchi the dog, with a human translating the barks.