It's pretty silly to suggest that the organization has "no sense of humor" and is only suing because the ad mocks them. If the ad were celebrating the event instead of mocking it, there is no question that they would respond in exactly the same way. In fact, I'm sure they would be much angrier - since this ad is parody, it clearly does not imply any endorsement by the Burning Man organization.
As this blog itself has discussed repeatedly in past years, the event organizers have no qualms about using IP law against anybody else who tries to profit off of Burning Man. (They even require that all attendees turn over their copyright for all pictures and video taken at the event. Or something like that. They used to, at least. It's a policy widely supported by the attendees themselves.)
I'm a regular burner and a big fan of the event, and I totally cracked up watching the ad - it actually reflects the complex feelings that many burners have about the event.
Ultimately, if the ad is just a comedy sketch shot on a studio set, the brief shots of BM IP are clearly fair use. But if the ad was shot at the event, by someone who had signed their copyright over to Burning Man, and then decided that the law didn't apply to them, I wouldn't be the least bit upset if a judge decides that this ad infringes on Burning Man's IP.
This isn't really "outside the box." Valve has been doing this for years with their games (Team Fortress 2, Left 4 Dead, etc.) If you cheat, you're banned from playing on any server that has the "anti-cheat" flag set, which is most of them... so you're stuck playing on a few servers mostly populated with other cheaters.
The only difference I can see is that, in Valve's system, non-cheaters are allowed to play with cheaters, if they want... you can even activate cheats, play on the the cheat-enabled servers, then disable the cheats and go back to the main servers again, because you only get banned for cheating on an anti-cheat server.
The website, like the press conference, is designed to confuse people, but only temporarily. This is where it differs fundamentally from the material that trademark law is designed to prevent -- the Yes Men WANT to be found out. That's how they get their message out. It doesn't do them any good to really trick people into thinking that the Chamber of Commerce has changed its stance on climate change; they want people to believe it only long enough to be outraged when they discover that the real Chamber hasn't actually done anything at all.
There's nothing in law to indicate whether this sort of use of trademarks is (or should be) protected, and (not that I'm a lawyer, but) I know of no case law on the subject.