Banding parrots is a way to be able to identify them, in case if they ever get lost. Some breeders band their birds to prevent incest, because it allows them to tell them apart.
However, there are some risks, as the band could get caught on something, and that can severely injure the bird. I've rarely ever heard of these cases though, so they may not happen that often.
If you don't feel comfortable about banding your birds, it's possible to get a microchip in as well, just like one would a dog or cat. It would be able to be scanned by a vet or rescue organization to identify the bird, just like how a band would. Might be a bit more expensive though...
Dudley's the only bird in my house that currently has a band on. Because my mom bred a lot of tiels, budgies, and finches, she has a band maker.
If you want to read more into legbands, here's an article on it.
http://www.beautyofbirds.com/legbands.htmlThere are both pros (tracking down original breeder, identification in case if bird is lost, etc.) and cons (risk of bird being injured, or broken leg because leg band got caught), so you have to decide yourself which is more important to consider. Keep in mind that microchipping is also an option, if you want a safer alternative. However, leg bands let people know right away that the bird belongs to someone, while microchips require scanning to see if there's one there at all, and since not everyone knows about the possibility of microchipping a bird, I don't see it to be as effective all the time.