If your parents do not want to get a dog because they work long hours, they must be convinced that you are not going to take care of him and her. What can you do to change their minds? Do you do chores around the house? Do they see you vacuuming the floor, cleaning the kitchen, cleaning the bathroom, etc.? What about your bedroom? If that is a real mess, and you do not even take care of your own space, why should they think you are going to go out of your way to take care of a new dog?
You need to discuss this with them even more. There is something else involved here, not just the amount of time they spend at work.