This is an interesting question to consider when thinking about this excellent short story. Of course, whilst the parents do face what they show fear about--the lions in the African scene that the children have created in their futuristic nursery--they do not choose to face this fear at all. Rather, their children manipulate and trick them into getting them into the nursery and then lock the door, ensuring their parents' death in the most horrid of ways.
Therefore, if I were you I would want to discuss the fear of the children in being taken away from the nursery which they have become so dependent on. Notice how Peter threatens his father when he suggests that they should have a "house-free existence":
"I don't want the nursery locked up," said Peter coldly. "Ever."
Also, when he makes this final decision to turn off all of the mechanical machines that have taken over their lives and had such a negative impact on their children, note Peter's reaction:
"Don't let them do it!" wailed Peter at the ceiling, as if he was talking to the house, the nursery. "Don't let Father kill everything." He turned to his father. "Oh, I hate you!"
However, in spite of this step in the right direction, the father crucially and fatally gives in, allowing his children one more play in the Nursery and thus sealing his own fate. So, when we think about facing fears in this excellent short story, the biggest fear is the children's fear of being without their nursery, which they never truly have to face.