Your dad is being selfish. He has it in his head that you are an extension of him and that he can tell you what to do. You are challenging this and its a threat to your dad's ego. Mom is just his peon in this. Why are you looking for his approval? If you are entertaining this delusion of his, you are participant in the dynamic. Here's a simple rebuttal: "I don't want to drive. I can go back to driving anytime I want to. This opportunity is now and I'm taking it. It's not your decision, it's mine." Now change the subject.  


I also think that they might be showing some cognitive limitations due to aging. If you pay attention to older people, they tend to do something that very young people do - they lazily categorize everything and then anything that doesn't fall into that neat compartment inspires cognitive dissonance and they get super reactive. Your father has neatly dropped you in to the "driver" box and can't reason his way out of it. Because this fits in his head, he will not let go of it until he finds a new box to put you in.  


You are being childish threatening to have a temper tantrum if the word is uttered. Learn to express and assert your boundaries without shouting. Listen and be attentive to your parents and you'll notice that they make poor counsel on certain subjects. Just nod and smile and use your own wits.  


TL;DR: you don't need to listen to your parents. Grow a spine and assert your boundaries without having a fit.