A Connecticut teenager at the center of a 16-month custody battle involving the Massachusetts child protection agency was allowed to spend Father’s Day at home without any state supervision, a sign that her parents may be on the brink of getting full custody of her. “She came home, and we did a barbecue,” Lou Pelletier, her father, said in a telephone interview Sunday night from the family’s West Hartford home. Pelletier said his wife, Linda, who has been staying at a hotel near Justina’s residential treatment center in Thompson, Conn., drove the 16-year-old to the family home early Sunday afternoon....