In Fahrenheit 451, what is the role of family in the characters' lives, particularly in relation to the TV parlor "families?"
To answer this question, take a look at the relationship between Mildred and the family from the parlor walls. For Mildred, this group of fictional characters constitutes her real family. She takes far more interest in their shows than in her husband, Montag. Mildred cannot even remember, for example, where she met Montag.
For Mildred, watching the family is about escapism. When she is involved in the parlor walls, she does not have to think about what is really happening in her life, and we know that she feels empty and depressed because she tries to kill herself early in Part One.
Similarly, this idea of escapism is further reinforced through Mildred's friends, Mrs. Bowles and Mrs. Phelps who come over to the house to watch the family. They show no emotion when they talk about their children and their husbands, preferring to focus on the parlor walls. It is only when Montag reads "Dover Beach" that the emptiness of their existence is made apparent.
In contrast, look at the relationship that Clarisse has with her family. They talk, they laugh, they spend time together and, as a result, are kept under surveillance by the government. This demonstrates that the state uses the parlor walls as a method of social control and this explains why so much emphasis is placed on watching the TV family, instead of cultivating relationships with real family.
The parlor TV walls are actually interactive. They talk to the person who has signed up to engaged with them. Montag and Mildred have three walls and Mildred longs for a fourth so she can have a bigger role in the lives of the families on her programs. The TV walls take the place of real family interaction and the parts played by the owners are minute and meaningless, but it gives them the illusion of being part of the TV families. The entire society in the book is based on giving its members the illusions of actual feelings and relationships so the TV walls play their role to a tee. Family and relationships in general cause people from time to time to become unhappy and dissatisfied. The TV families can step in and give people the illusion of being part of a family without any of the pain or the joy that can come with actual relationships and interactions.