When it bites our hand, we may drop the bird to the ground, some even strike their bird to stop this behavior. Being chased, falling to the ground, or being struck may be something these birds will work to avoid in the future, maybe not. An animal trained with positive reinforcers will look forward to these interactions and work towards earning the reward. Parrots in the wild congregate and spend time screaming for many different reasons. You may be positively reinforcing the scream, not the message you are trying to get across.
Punishment is a method that has little use in bird training.
Punishment always decreases the probability of an act occurring again. Punishment takes place after the action, is not directly associated with the action, and the animal cannot stop the behavior to avoid the punishment. A hole has been chewed into the edge of this beautiful table, which took place probably hours before. You sweep the bird up, toss it in the cage and cover it up. This is punishment.