Well, I'm in Illinois, and as far as I can tell, there is no law requiring that one be 18 to own a car.
That said, the reason car sellers require parents to cosign for buyers who are under age 18 has do with contract law.
A "contract" is simply a court-enforceable agreement. Certain contracts are "void." For example, a contract to sell marijuana would not be enforced by a court on the ground that it is "void" as a matter of public policy. Other contracts are "voidable," such as those entered into by anyone under the age 18. In other words, a minor can void a contract at any time and get his/her money back.
So, to answer your question, yes you do need a parent to cosign if you are under 18. It has nothing to with "having the money" and everything to do with the fact that you could walk in and demand a refund at any time until you turn 18!