Parking tickets are not moving violations, and do not directly affect your car insurance. Since the infraction did not happen while the car was traveling the roadways, it is not regarded as being as serious an offense as speeding or other tickets you get while operating the vehicle. The problem here is that not paying the ticket introduces new legal issues that will affect your driving privilege and your car insurance rates.
When you do not pay a parking ticket, the state will suspend your driver's license until the ticket has been paid. Once paid, you will still have to pay a fee to reinstate your driving privilege. However, since you cannot legally drive while your license is suspended, your car insurance will be canceled as well. Even if your insurance company does not find out, if you get into an accident they will deny the claim on the grounds that you were not a legal driver.
Parking tickets do not directly affect your insurance rates, but a suspension of your license or registration would result in immediate cancellation of coverage. If the vehicle is required for the use of people other than yourself, you will have to make special, sometimes complicated, changes to the vehicle registration in order to allow another person to purchase insurance on the car. Insurance is purchased on the car, not the person, but only a licensed driver can be authorized by the insurance company to use the vehicle.
Another thing to keep in mind is that tickets received in another state can still affect you. Most states are connected to a national database of drivers, and the tickets you receive in one state will count against your license in another one. Similarly, if your license is suspended in one state, you cannot move to another state and get a new license until you have had the suspension removed from the first state. Since buying insurance is contingent on being a licensed driver, you will not be able to get insurance until you have your licensed straightened out.