When a third party injects him or herself into an existing dispute between a plaintiff and defendant?

That third party will gain the status as intervenor

Explanation:

The third party could obtain the status even without the consent of the initial two parties. The only thing that the party need to do is to convince the court that they are heavily influenced by the case between the first and second party. In most cases, Intervenor tend to appear on appellate proceedings. It rarely occurs in higher court level.