The child should buy their own car as well as pay for their college. But it seems that's not the way it works in todays day and age, many kids feel (and parents too) that it's the parent's responsibility, kids want everything handed to them, they don't want to work for it, but that's a whole different thread.
If the parents agree to buy the car, who should pay? I think it depends on the custody arrangement. I can give you an example of my sister. My sisters ex lives in CA, she lives in PA. My niece (16) goes to her dad's in the summer for about 2-3 weeks, that is the only time she sees her Dad. He does pay child support (has never missed a payment) and pays quite a bit. But things he also does....sends extra $ for xmas, school clothes, pays for all her instruments (band), paid for her braces, he did buy her a car and had it shipped to PA. I think he figures that my sister does all the child rearing basically and it's the least he can do. He has a college fund set up for her as well. I think if he lived closer to them and was more involved in the raising of her, then he probably wouldn't do all the xtra stuff. Plus, he is finanacially able to do all the xtra things. My sister never has to ask him for anything, he just does it.