A: No. Brain Parasites don’t inhibit their host Keeper’s power, but in the case of the Doctor, he cannot treat himself. As the card says, he can only discard Brain Parasites if attached to one of your other cards.” Of course, if you have the promo card The Android Doctor, then he can cure the Doctor of Brain Parasites, and vice versa (the Doctor could cure the Robo-Doc). Like the Doctor himself, however, the Robo-Doc can’t cure himself either.
Here’s the full thematic explanation of why Brain Parasites don’t inhibit Keepers’ abilities, but they DO keep the Doctor from healing himself:
The reason is not due to the fact that he’s lost his curative powers. If you have Brain Parasites, then your every action will be subject to the whims of these aliens who have invaded your head. The Brain Parasites might WANT the Engineer to repair that computer. They might WANT the Captain to steal the Scientist. After all, they don’t want to do stuff that’s bad for their host, right?
But it simply doesn’t make sense for them to ever allow the Doctor to use his curative powers to evict themselves from his own cozy cranium. In fact, if there were multiple Brain Parasite Creepers in the game, the Brain Parasites would, logically, bar the Doc from curing anyone of Brain Parasites (which would effectively nullify the Doctor, since in this game that’s the only curable disease in known space, apparently.)
PS: The real question in my mind is why the Brain Parasites don’t nullify the powers of The Expendable Crewman, since his “power” is that he frequently gets himself kidnapped or killed. Kidnapped is fine… I can totally imagine a scenario where the Brain Parasite might want it’s host to get stolen and taken to another ship (“Oh goody, we shall spread to a whole new crew!”). But one of the possible (though perhaps drastic) ways to get rid of an attaching Creeper is to destroy the host Keeper, and the Expendable Crewman’s power is to somehow step into the line of fire, even when he’s not the intended target. Maybe the “power” of the Expendable Crewman is just so strong that even having his mind controlled by an alien being can’t keep him from being a hot-headed klutz.