Again, he could just give her the cable, I mean is it that important that you have full batteries?
Think of it like this, what if you were the one with a broken laptop cable, would you hit her if she didn't give it to you? Maybe you would, but in the end what would that accomplish? In fact you would probably be in the same spot you are now.
Also i have a feeling that you probably hit her back, or that you are not telling us something, if your parents are dumb enough to believe her, and she actually is lying then they would be dumb enough for you to put pillows in the bed and sneak out the window.