It is difficult to say who the protagonist is in this play. The Director is at the mercy of being caught between actual characters and actors on stage. He must negotiate between the two groups what the play will be. However, he never fully grasps how "real" the characters are. He is one of the main characters in the play. He has a prominent role. But he, the actors, and the stage hands seem to be more like audience members reacting to the behavior of the six characters. However, you could make an argument that the Director is the main protagonist because he is the "author" they are in search of. They need his stage and his direction in order to live their character lives. Without him, do the six characters even show up or emerge in the theater?
On the other hand, the six characters are all protagonists. The Mother, Father, and Stepdaughter are clearly the most prominent of the six. Each of these three is a protagonist in his/her own right. And each of the three antagonizes the other two in some way. So, you could argue that they are antagonists to each other. And you could clearly argue that the Actors are the antagonists to the six characters.
The protagonist is not always necessarily the "good guy" or the "good girl." It is more broadly defined as the main character(s). You could make the argument that it is the Stepdaughter. She is intent on exposing her father's alleged sin and she wants the play to be true. Much of the play and the play-within-the-play follows her monologues. The father also wants to tell his side of the story and as spokesperson for a lot of backstory, one could argue that he is the main character. The mother is pivotal in the story because the story largely follows her life. She was married to the Father, they had a Son, and then the Father left her. His son goes to live somewhere else. The Mother takes up with the Father's secretary and these two have three children: the Stepdaughter, the Boy, and the Girl. Every bit of the drama occurs because of what happens to the mother.
In the overall play, you could pick any of these four (Director, Stepdaughter, Mother, or Father) for the reasons stated above. In the play-within-the-play (that of the lives of the six characters), you could argue for Mother, Father, or Stepdaughter. This is one of the most original plays ever written, so it doesn't conform to normal or typical kinds of analysis. This makes it difficult to pick a single protagonist.