You don’t need permission for true parody but changing the lyrics (unless you do so to comment on the original work or author) isn’t that.
Take Amish Paradise. It commented a bunch on the Amish. But it didn’t say anything about Coolio or Coolio’s work.
Webster offers a lay definition not a legal definition. Often in law words are interpreted to have meanings different than they normally would. For example a company would be considered to be a person for the purposes of a law saying “No person shall dump oil in the river”.