The parody is invoked when the atheist wishes to illustrate their position by attempting to place the theist in their shoes. Most of the attributes of the FSM or IPU are essentially arbitrary. The two necessary attributes of the parody are:
1) The theist will consider their to be no evidence for the existence of the parody
2) The theist will not consider the parody to exist
The weird, self contradictory and humorous attributes of these parodies are there, essentially, to ensure that these two key attributes are preserved for the vast majority of people.
The process goes like this:
1) The theist acknowledges that the atheist believes their is no evidence for the existence of God
2) The theist acknowledges that the atheist does not believe in God
3) The theist asks the atheist whether he considers argument X to provide a reason for the atheist to believe in God
4) The atheist asks the theist whether he considers argument X to be applicable to the parody as well
5) If the theist agrees that argument X applies to both God and the parody yet does not wish to believe in the existence of the parody then he must accept that from the perspective of the atheist, argument X is a poor argument for the existence of God
6) If the theist disagrees that argument X applies to both God and the parody then they are enlightened as to the specific reason why the atheist cannot accept their argument. The theist is now the prime position to restate their argument in order to clear up the difficulty which the atheist is having.
Unfortunately, this process rarely happens because theists often find such parodies to be offensive and dismiss the atheist when they bring them up. I feel that this is largely justified because the FSM is, afterall, a parody, and it shouldn't be up to the theist to discern whether he is speaking to a militant or amicable atheist especially on the internet where these things are often hard to determine.
In the spirit of making the argument more workable, I hope I have explained here quite clearly what the atheist is attempting to do when he invokes the FSM or IPU etc and would ask those theists who find such parodies to be offensive to please offer an alternative which they would not find offensive but clearly displays the attributes necessary for the parody to be valuable. After all, what is the point of trying to initiate this process in the spirit of furthering understanding when we know that half the time, it will turn into a flame war due to our inability to find an unoffensive parody.