No - you don't have to kill the dog. The law makes exceptions for these things. And even if I did, I wouldn't - I'd pay a fine. There's a trick to getting dogs to move away - just honk the horn loudly and drive slowly, ever so slowly away, whilst picking up speed gradually. Dogs have pretty fast reaction times (well, most of the dogs I've had have had good reaction times), so they'll move out of the way.
But yeah - don't kill the dog unnecessarily. I'm not a vegan, vegetarian or PETA person, but if you don't have to kill something, then don't. I understand what your Dad is saying, and I agree that a Dog's life isn't worth as much as a Human's life, but a Dog can still feel pain and I think that saving the dog's life is more than worth the fine.