Thanks for your help, guys. I actually posted this on a legal forum, Free Advice Forums. Here's the response:
Your work differs in that you can be sued for copyright infringement, whereas with an original work you can't.
Although parodies often fall under the "fair use" provision of the Copyright Act, whether a use is fair or not is ultimately determined by a court. Courts look at the purpose and character of the use (whether it is for a commercial purpose or a non-profit purpose or an educational purpose, for instance), and courts look at the nature of the work, and the amount and substantiality of the portion used in relation to the copyrighted work as a whole, and they look at the effect on the market for the copyrighted work and how it may affect the value of the copyrighted work, and they can also make a subjective judgment and base their decision on whether they like the parody or not. When more than a few lines of a song lyric are parodied, it is often judged not a fair use of the copyrighted material.
People who parody songs on a regular basis, like Weird Al, will usually get permission from the copyright owner prior to releasing their parody. Lawsuits often occur when the copyright owner hears the parody for the first time and doesn't like it, or starts to lose money because of it.
So it seems the best way is to ask the copyright holder. Now how to find that out...the label?
Just thought I would post the response I got so that if anyone else had a similar question, they could refrence this.