Bacteria can cross over, but few parasites can do so. Virii are another story--mutations aren't unheard of, but are certainly uncommon. Most transmissions revolve around open wounds in the water and ingestion. Mycobacterium and TB are the 2 most commonly discussed ones. Mycobacterium are everywhere, and seldom cause a problem, but the infections from them are very nasty and hard to cure. Infections of this sort are always the result of an open wound in the tank. While I have heard often that TB is transmissable, I have not heard of anyone who actually contracted it, so do not no the mechanism.
Overall, you're still safer playing in your aquarium than driving your car.
Overall, you're still safer playing in your aquarium than driving your car.