Most of the content of BJAODN is pretty clearly a parody of normal Wikipedia content. It has a lot in common with the Uncyclopedia (though frankly Uncyclopedia is amazing).
Firstly, I think to be a parody it has to be intentional. A lot of BJAODN isn't intentionally a parody. Secondly, that only means the authors of the BJAODN can use Wikipedia content under fair use, it doesn't mean other people can use the BJAODN under fair use. Fair use protects parodies, it doesn't allow people to use someone else's parody.