First of all, the illuminatti aren't trying to "take over" anything. That's not what they do. Secondly, they don't know who each other are, for the most part, and so don't have any single identifiable agenda. The illuminatti isn't an organization with a list of members and a plan. It's more a phenomena that occurs spontaneously within other secretive organizations, like the Masons or the Catholic Church or the C.I.A. with tiny individual cells that are acting on their own, and toward their own selfish goals.
Generally, the way one "joins" the illuminatti is that you are given clues by them, that lead you to think you are discovering them and some plot that they're working on, and because you see some advantage in it for yourself, you end up threatening to expose them if they don't let you "in" on it. But in reality, you have been chosen by them, and then played, so that you would think you had the upper hand, and knew what was going on, and had made them accept you. In reality, they are all lying to and manipulating each other in pursuit of their own goals, and none of them knows who all of them are.
The illuminatti are basically a bunch of self-serving anarchists who enjoy infiltrating "secret organizations" and screwing with them from within. The more secretive and the more "organized", the better fun they can have messing with them covertly. We can't say that some illuminatti somewhere doesn't have the agenda of advancing gay parenthood, because any one of them could be working toward any goal at any time. But it's safe to say that this would not be the goal of the illuminatti as a whole. They have no goal as a whole except to screw with the goals of whatever secret organization they are hiding within.