A final note: This has nothing to do with free speech and everything to do with copyright; if the movie is a copyright violation, then it is illegal. If it is not a copyright violation, then there is nothing wrong with it. There is no Free Speech fair use exclusion. Such an exclusion would make a mockery of copyright law, as nearly anything can be cast as free speech.
You may desire that it be OK to produce that movie, you may think it is morally necessary to allow people to produce such movies with copyrighted materials... I have no problems with those positions, those are perfectly acceptable normative statements. However, the issue of what IP law really says is not changed by such things, and confusing desires with reality does not enhance credibility.
If it's not a parody (which I certainly don't think it "clearly" is) then it's just out-and-out copyright infringement, which I will not stand behind and recommend that you do not either. We all need the basic principles of copyright to protect us from being exploited by large companies, and we should not weaken them for our own amusement.
Further question: Is the movie really a parody? If so, exactly what is it parodying? The "Stormtroopers" bit at the end may have the nature of a parody, but the rest strikes not so much as a parody as just an attempt at humor. If you sit and analyse anything long enough, you can ascribe nearly any attribute to it, even being a "parody" of something. I'm just plain not certain that the spot parodies anything deliberately; it would probably take a court to sort it out (though I'd personally be satisfied to talk to the authors for a bit).
Perhaps the movie is legit and AP was hasty in ordering it removed... certainly Dave's suggestion that AP should make friends instead of enemies is not out of line... but the question is not one-sided and we do not benefit from acting as if AP is totally incorrect.
In other words, yes, I'm directly contradicting Dave's call for people to mirror the file. Don't do it, it's not worth the fight. (Convincing AP that it's OK for people to do this is another issue entirely; copyrights don't need to be defended and aren't lost if you don't crack down on abuse. The movie didn't hurt AP whatsoever.)
We should pick our fights carefully, and this is not one worth fighting. If we dismiss the rights of others, even large companies, whenever it suits us, then how can we, the common folk, expect any better treatment from them when they decide to, say, download people's resume's from the internet and sell them for profit without permission from the resume owner?
As shown by the example above, it can often be difficult to determine whether a particular use qualifies as a fair use. It most cases, a copyright attorney should be consulted before undertaking any significant activity which would rely on the fair use doctrine as a defense to copyright infringement.
In reality, fair use is a complicated topic that essentially must be decided on a case-by-case basis in court. For instance, from the Bitlaw entry on fair use:
As an example of one of the misunderstandings, one of the clauses of the fair use exclusions allows you to reproduce a limited portion of a work ('substantially less then the whole') for the purposes of critique or review. The general public tends to believe that they are allowed to reproduce a large portion of the work for any purpose whatsoever... which is just not true.