The theory that Lizzie Borden was in some way abused by her parents is popular, because it would explain why a quiet librarian might suddenly turn on her parents and kill them violently with an axe, and trigger a court case that would rock 19th century Massachusetts.
However, since she was acquitted anyway.. It's all just speculation!
But... If Lizzie really did kill her parents, then the motive was more likely to be money.
Lizzie was enraged by years of living under her father's miserly ways (the family was very wealthy, yet didn't spend very much money).
She longed for a more luxurious and prosperous life, and she felt her parents were getting in the way.
The last straw was probably the notion that her dad might pass away leaving his money to his wife: Lizzie's stepmother.
She suspected if this happened, she would never live the way she wanted to, and this injustice was probably the motive.