It will depend upon state laws if a 17 year old (or any aged minor) can purchase a car from a private seller with cash without obtaining their parent's consent. The age at which a person can legally have ownership of a vehicle varies from state to state.
In general, a minor or juvenile cannot enter into a contract so thus cannot be a sole owner of a motor vehicle. A minor cannot normally actually own property. His or her parents technically own that property. If a parent will co-sign on a loan for a car, the juvenile can buy a car, but the parent will be the legal owner.
There can be exceptions to this generalization depending again upon state laws. For example Arizona has an emancipation statute for minors over 16 who can show they are sufficiently mature to handle their own financial and health-related affairs. Here a minor to petition the court for an order of emancipation. If the order is granted, the minor would then be able to enter into legally-binding contracts for the purchase of property, including cars.
Even those states that will permit a teenager to register a car will normally require a parent or guardian to sign on any legal documents or contracts since they are still a minor. The Ohio Bureau of Motor Vehicles (BMV) states if you are under 18 and want to title a vehicle, your parent or legal guardian must complete a minor consent form. A parent or legal guardian must also accompany you when you appear in front of the Clerk of Courts title office staff to have a vehicle titled in your name in Ohio if you are a minor.
In Michigan there is no age requirement for titling a vehicle in Michigan and no minimum age for obtaining insurance. However it is unlawful to knowingly sell a motor vehicle to an unemancipated minor without written permission from a parent or guardian. The seller must have a consent form completed by the parent or guardian of the minor at the time of the sell. By law, the seller is required to keep this consent form for 3 years.
The form needed to be filled out is titled Parental Consent to Sell a Vehicle to a Minor. It states right on this form that Michigan law requires an owner to obtain parental permission before selling a motor vehicle to an unemancipated person under the age of 18. The seller may comply by having one parent or the legal guardian complete this form. The form also comes with a warning stating that this form does not change the contractual liabilities with respect to the sale of a motor vehicle to a minor.
In most states if you are 17, and thus still a minor, you will also need a parent or guardian to sign on any legal documents such as a title to a car or an insurance policy. Most insurance companies will not allow you to sign by yourself on a policy until you are 18 and considered an adult. So the insurance policy may be in your own name at 17 but with the consent of an adult since they may be held liable for your actions as a minor if you fail to fulfill your obligations.
To find out for certain what your state's laws are regarding the minimum age requirements for owning a vehicle check with your local Department of Motor Vehicles or like state agency. Until you find out about the laws of your state it would be unwise to buy a car as a minor without consent or if you are a private seller to sell a car to a 17 year old juvenile since you could be breaking the law if you do not have parental consent for the transaction to take place.