They can get sick. I think there are certain kinds of avian flus but it's so unlikely for your bird to be able to catch that. I don't think they can get a cold but they can get sick. However, if the parrot is eating, pooping, vocalizing, playing, and weighing normally, I don't think it is sick. It could just be imitating noises or even copying you. It's not unheard of parrots catching sneezes and coughs from their owners without being sick at all
