One of the problems is that birds do things for a reason. They don't always do what we want them to do, and it does no good to hope that they will change just because we want them to change. It can be hard to understand why parent birds would do something like this. Shouldn't all parent birds naturally love and protect their chicks? But as you know, not even all humans are kind to their offspring, and birds can be much like humans.
In the wild, a sick bird is a danger to the flock. A sick bird might transmit its illness to the others. If it flies poorly or acts odd, it might attract the attention of predators who will catch any bird in the flock they can, not just the sick one. For this reason sick birds do their best to conceal the fact from the others - they know instinctively that losing the flock, being left behind, means death.
It is difficult to say why the parents started picking on the one baby. Maybe it was injured by the other baby, or stepped on accidentally. For some reason, when the parents saw the injured baby they decided to get rid of it. This can be necessary in the wild, because the other baby could become ill. In the wild it is difficult to raise chicks - so much food to bring and so much energy to spend on care - so tending to a sick chick could take enough food and time away from the healthy one so that both die. Or, perhaps there is something not right mentally with the parents. They both have a harmful mutation, and perhaps there are invisible problems that make them...well, let's say insane for lack of a better word, or at least unfit to be parents.
It will do no good to keep putting the injured baby back with them in the hope that they will be good parents all if a sudden. No, they have decided that this chick should die, and it doesn't matter what anyone else thinks. It presents you with a difficult problem.
The only way for the chick to have any chance is if you can do two things. One, the wound must be cleaned and protected and disinfected, and it still might not heal properly. A vet could do the best job at this. If there is no possibility to get to a vet, you can try to flush the wound with clean water for a long time to try to wash away dirt and germs. Many doctors now say not to use peroxide, as it damages healthy tissue. An antibiotic ointment might help prevent infection. There are some "artificial skin" products that form a seal over wounds to protect them - but usually these have some solvent or alcohol and hurt like crazy when you apply them.
Second, for the bird to live you must feed him, and as you have learned this isn't easy. The food must be correct, such as baby parrot formula, and it must be not too hot or not too cold. Too hot and it can burn an actual hole in the tender crop, or cause it to form a sore or wound. Too cold, and the food can congeal in the crop like cement so that no food at all makes it into the digestive system and the bird starves. The bird may refuse to eat from an unfamiliar device. Even with th correct food and technique, it takes a lot of time. Babies must eat every few hours, and they must be kept warm. So you see it is a daunting task, to try to save this little life.
And even if you do save him, h might not be a healthy bird because his parents have genetic problems. He should never be allowed to reproduce and pass along bad genes. It is not a good prospect. I hope you can find a vet to help you, because the baby is in a lot of pain. Birds are very sensitive, and that is a big wound, and it probably hurts. A vet migh be able to give the bird some medicine to help with the pain. Sadly, and I sorrow to say this, the baby may have such a hard time ahead of it that it would be kinder to stop the suffering instead of dragging it out. A vet could advise you on the baby's chances, and perhaps there is a positive outcome that medical help could bring. All you can do now is to think as hard as you can about what can be done, with compassion and all the wisdom in your heart, and do what seems right and best to you. That is all any of us can do.
It may be too late to help this little baby, but you can take some steps to prevent this from happening again. The parents will probably never be good parents, and if they have more chicks the same thing may happen again, and both you and the future chicks will sorrow over it. Plus, the parents have bad genetics that should not be passed on. You can find information here, I am not a bird breeder so can't really advise, but probably the males and females should be kept in separate cages from now on. That is probably the only way to keep them both and to prevent more chicks.
Many people have birds that lay eggs and raise chicks without problems, but you have been very unlucky. There is a lot to learn in a very short span of time, and it might not be enough to save the baby. I'm very sorry. Do the best you can, ak for advice if you are not sure, and do your best to prevent it from happening again. That's all anyone can do. I hope things will work out. Thank you for caring enough about the baby to come ask a bunch of strangers for advice, and don't stop loving the wonderful feathered beings in the cage.
