I'm currently on my parents insurance policy and I am in the market for a new car. If I finance the car in my name will my parents policy still cover the car? What if I finance it in my name with one of my parents as a cosigner?

In order for it to be on their policy, they have to have what the insurance company calls "an insurable interest", aka they have something to lose if you wreck it.

What this means in normal-speak is that they'd have to cosign, if you want to be on their policy. You may want to check out how much the same insurer would charge you if you were on your own.