Nature protects a species, flock or family from being attacked by a predator tracking an individual member that is sick or injured. The parents are protecting the rest of the family by removing the injured cygnet from the area.
You might contact a local rehabber or wildlife rescuer to see if they may help in rescuing, treating and relocating the injured cygnet. If the cygnet is continually chased, it may become so stressed that the cygnet succumbs to dehydration, hunger or predation. Ultimately, it may die.
Now, having said this, you need to be extremely cautious in contacting a wildlife facility. Federal and state wildlife officials are trying to kill all Mute Swans in the U.S. so that the larger Trumpeter Swans can be introduced for Trophy Waterfowl hunting purposes. If you live in a state where wildlife officials require wildlife facilities or veterinarians to kill the cygnet if it is taken to them, then they may also try to kill the entire family. The cygnet does need help, but you must first determine if anyone contacted will indeed help. You may have to contact several wildlife rehabilitation facilities before you get the answer. Animal control or state wildlife officials should not be contacted. The Regal Swan