Maybe no one knows. I read on the Wiki that they're everywhere now on the lower planes and not tied to a particular faction, but that doesn't help me much in formulating a theory.

Below contains a couple of in-game spoilers and wild guessing about what's going on.

My theory at the moment is that the dreams the PC has are not caused by the tadpole at all, but by a succubus/incubus. This would have to be a super special tadpole to seduce its host into accepting it when it could just eat the brain and have control. But a succubus's MO is seduction, and those dreams are most definitely seduction. So I'm trying to decide if the PC is a pawn for one side or a tug of war rope between both. Maybe I'm way off base, but a figure is standing over the PC the first time they sleep after using the tadpole, and I'm pretty sure it's a winged figure. Then, if you keep using the tadpole, you get a vision of looking over a city (I assume Baldur's Gate) that is half in flames. What purpose would that serve for the illithids? Guess I'll keep wondering for now.