You've got the law on your side, I believe, from my two quarters of Intellectual Property Law that I studied.
Note: My comments are not to be construed as legal counsel, yada, yada.*
Parody is "fair use" and creaters of parody are protected from most copyright infringement. But it is important that your work meet four parts to stay within the definition of "parody."
Section 107 The Copyright Act of 1976 outlines what a court has to consider when determining if something is fair use:
1) Purpose and character of the work
2) Nature of the work
3) Amount and substantiality of the portion used in new work
4) Effect of the use upon the potential market for or value of the copyrighted work
So, I would say that a fan film is NOT a parody, because a fan film is an homage to the work in question: Fan films generally use the established characters in established situations. The idea of a fan film, usually, is that the creators want to continue the franchise that was dropped by the network, or they want to record their cosplay as their favorite characters. This fails the “purpose and character of the work” and “nature of the work” tests.
Parody includes commentary on the work being looked at, or social commentary. The Twisted Tune™, “What if God smoked cannabis?” is social commentary, so it is protected. Weird Al’s Star Wars songs are commentary on the films, using clever twists of the lyrics of the songs he borrows from, so they are parody. Weird Al’s “Like a Virgin,” and “Eat It!” are commentaries of the songs and videos themselves, so, again, protected.
So, if your film is a commentary on the Dr. Who franchise and the typical actions, behaviors and situations one can expect to see in a Dr. Who franchise, then it is a parody. If it is just a case of “If I were Dr. Who, I’d do this…” then it’s a fan film.
Here's a link to a summary of the idea of "fair use."
http://www.legalzoom.com/intellectual-property-rights/copyrights/2-live-...
*See? A parody of typical legal disclaimers.
As I said, it's an obvious parody. We are satirizing the show itself, not just doing a Doctor Who with jokes (as you pointed out, Doctor Who has jokes itself). We've talked about having the actor playing The Doctor use a Scottish accent to play on David Tennant being Scottish and we've talked about maybe doing it Australian, to play on George Lazenby (James Bond in On Her Majesty's Secret Service, thereby zinging not just DW, but subtly zinging British pop culture as a whole). It is also a parody of fanboys, as the villians (based on the Autons) are his collection of action figures come to life. There is also a joke that parodies the TARDIS redesigining its control room whenever it wants. It's an obvious SNL style parody. His argument is that just because it's a parody doesn't make it not a fan film. I say he's wrong. Being a parody, not a straight short film, absolutely makes it not a fan film.
One lab accident away from being a supervillain! Bazinga!