One final note:
In the United States, parody is protected by the First Amendment as a form of expression. However, since parodies rely heavily on the original work, parodists rely on the fair use exception to combat claims of copyright infringement. The fair use exception is governed by the factors enumerated in section 107 of the Copyright Act: (1) the purpose and character of the use; (2) the nature of the original work; (3) the amount and substantiality of the original work used; and (4) the effect on the market value of the original work. Generally, courts are more likely to find that a parody qualifies as fair use if its purpose is to serve as a social commentary and not for purely commercial gain.
Parody | Cornell Law School Legal Information Institute
This means that you are protected by the law when using parody, as long as you are using it as social commentary and not purely to make money off of it (although many do!).
Writing a parody can be a lot of fun. Good luck with yours, and please feel free to share it with us!