It depends on which parasite you're talking about. Suppose it's fleas, for sure. Any of the animals, including people, can get fleas on them. Ticks can spread from cats to dogs to people. Heartworms are generally transmitted from dog to dog, and it's always through a mosquito. So one dog can't give another dog heartworms unless, of course, it went through a mosquito first. Ringworm can be transmitted from one to the next. Mites (like ear mites) can go from one pet to the next. So it's vital that we figure this out right away, so we don't spread it throughout the family.