It sounds to me like these birds are not going to get along. For their safety, they should not be allowed out at the same time. This is going to escalate to a serious injury or worse. Parrots generally do not do well in odd numbers. It is interesting that the aggressor is the third bird that was brought in. In most cases, two birds will bond and bully the outcast. In the wild, birds really do not have friends. They may flock together, but split into pairs for breeding and once bonded they do not have close physical contact with other adult birds. The flocking behavior is for protection, but within the flock, the pairs stay together. Another thing here is you have birds from three different genus. And actually, just for fun you should research your Green Conure because the taxonomy, or scientific classification has changed. They are no longer classified as a Conure, but rather one of the Neotropical parakeets. They are more closely related to the macaws. For you, this is mainly a point of interest. As far as the fighting issue, there isn’t much you can do about this other than to keep them separate for their own protection. I realize this may mean re-homing the newer bird if you are not willing to change the open cage policy. Even birds of the same species will have an instant dislike for each other for no apparent reason. Like people, birds have their own ideas of who they like or dislike.
Thank you for asking Lafeber,