Your dad telling the truth or lying will be revealed soon enough. You can only lie for so long. If your mom isn't going to tell, then I don't think I would. Is hepatitis caused by drinking? Maybe the doctor was going to tell him he needed to quit drinking?
I suppose I'd call. But I'd call knowing it probably wasn't going to make a bit of difference with my dad. What if the doctor told him he couldn't drink anymore, think he'd listen?
Why doesn't your mom tell the doctor?
I hope he'll be okay