Who is Peter Billingsley's mother Is it Barbara Billingsley of leave it to Beaver?

No, Peter Billingsley's mother is Gail Billingsley, not Barbara Billingsley. Barbara Billingsley was an actress best known for her role as June Cleaver on the TV show "Leave It to Beaver." Peter Billingsley is an actor and director, known for his role as Ralphie in the movie "A Christmas Story."