You should not have the new bird in the cage with the pair. It is never a good idea to have an odd number of birds in a cage, and when you are trying to get a pair to breed, there should only be one pair in the cage. They might attack and kill the new bird because the cage is their territory. You said you rescued it so I’m not sure if it came from someone else or it was flying loose outdoors. Either way, when you bring a new bird home it should be kept separate from your other birds for 30-45 days in case it has any diseases. If the bird was found outdoors, it would have parasites and spread those to your birds. If it was outdoors, you should take it to a Vet to make sure it doesn’t have any internal or external parasites.
As to the pair, if the male was already bonded to the other female, he may not bond with the new bird right away. Some birds will never bond and breed. In the wild they get to choose a mate, and they do not always accept the mate we choose for them. So they may need more time to bond. Signs of being bonded are spending all of their time together, feeding each other, grooming each other and mating. They need to be on a nutritionally balanced diet like pellets, and not just a seed mix. You should also offer dark leafy greens and chopped veggies. When they are set up to breed with a nest box, you should offer an egg food daily – cooked egg with the shell washed, crushed and cooked with the egg. The female especially needs this because it takes a lot of protein and calcium for her to produce eggs. The nest box should be attached to the outside of the cage as high as possible. There should only be one pair of birds in the cage. They need a lot of privacy so the cage should be in a quiet place where they only see you when you have to feed them or clean the cage. They should not be able to see other birds. The nest box should be removed after each clutch, whether they hatch chicks or not. She should only be allowed to breed and lay eggs twice a year, with the nest box removed for 6 months after each clutch. All you can do is provide what they need and give them plenty of privacy and hope they breed.
Thank you for asking Lafeber,