In reality, Elizabeth Parris, Betty's mother was still alive throughout the trials. However, Miller chooses to portray Rev. Parris as a widower perhaps to explain Betty's freedom to run around with Abigail and the older girls and participate in their schemes.
In real life, Elizabeth Parris was an invalid, and Betty was a sickly child. She often was confined to bed or prevented from doing household chores, so she would have been especially interested in hearing Tituba's exciting tales as a diversion.
In The Crucible Miller does not mention the cause of Elizabeth Parris's death; he simply describes Parris as a widower in his opening notes. If Miller just intends for the audience to believe that Elizabeth died from her chronic illness, this would have frightened Betty because, as a bed-ridden little girl, she would most likely think that she was going to endure a life of illness and an early death as her mother had.