What symbols does Billie Jo use for her mother, her father, and herself? How do they represent each character's personality?

To answer this question, we must look at Billie Jo, her father (Bayard), and her mother (Polly) individually.
Billie Jo: At the beginning of the novel, Billie Jo is a 14-year-old girl with a talent for music. She was taught by her mother when she was 5 years old, and in her opinion, nothing compares to it. The first symbol, music, is a clear indication that Billie Jo is highly creative, and this can be seen through her aptitude and talent for it. She also loves apples, which have traditionally been a symbol of knowledge and wisdom. Billie Jo is wise beyond her years, having endured hardship from childhood, and her natural curiosity is what feeds her knowledge. Fire, and the resulting burns on Billie Jo's hands, is a symbol of destruction (she is unable to play the piano afterward without experiencing great pain), resurrection (she begins to heal her relationship with her father after her mother dies), and the cycle of death and life.
Daddy (Bayard Kelby): Daddy is a wheat farmer, struggling to keep his farm afloat in the midst of the Dust Bowl. Although he has an argument with Ma about the crops he should be growing, he insists on wheat. Wheat can symbolize many different (and at times ironic) things: abundance, life, fertility, rebirth, and resurrection. The theme of resurrection is more evident in Bayard after his wife dies: he cuts Billie Jo off, which makes her run away, but after she calls him from an Arizona train station and tells him she's coming home, he begins to take more of an interest in her. Bayard also contracts skin cancer, another important symbol. Cancer seems to be a representation of anger, resentment, and guilt, all of which are evident after Ma and the baby die.
Ma (Polly Kelby): Ma doesn't get much character development, but she is a hard worker and keeps their house clean despite the volumes of dust in and around it. The symbolism of the house is both physical and psychological: it can evoke parallels with sacrifice (it is hinted that Ma was brought up well, but gave it up when she married Billie Jo's father), endurance (she keeps the house clean and neat, despite the rigors of a hardscrabble life and a failing farm), and masking true feelings (it is evident that Ma is slightly resentful and bitter that Billie Jo gets opportunities that she never had). Ma also grows apple trees, potential symbols of death (she dies in childbirth) and immortality (Billie Jo mentions her throughout the novel after her death). Ma shares the symbolism of fire and music with Billie Jo as well—fire as a representation of life and death, and music as a symbol of her creativity.