First, being a parent, listen to your father! He probably knows more than you think!

Second, being a rider. Sit on your ATV, pick your feet up. What happens? Now, find a dirt bike, sit on it, pick your feet up. You fall over right? That's about how much more dangerous a dirt bike is over an ATV.

Given the same power/suspension/brake levels, you can accelerate, brake, turn and crash about the same on both. The ATV does give more stability mostly because you have more wheels on the ground.