Whether you can keep an uninsured vehicle parked on a public street will depend upon your state's laws. Typically, a state requires vehicles to be registered and therefore insured. If a vehicle is not registered than you can take insurance off of it but in general state laws will not permit that vehicle to be parked on a public street or roadway.
It may be possible to keep your car parked or stored without insurance on it but usually you must on private property like a driveway or garage. In many states if law enforcement finds a car parked on the street that is without registration or expired registration they can cite it or may even be able to have the car towed.
If your car is inoperable you may be able to get special status on it from your state's Department of Motor Vehicles so that you could suspend registration and take insurance off of the car. Again state's laws differ so you would have to check with your local DMV to see what they allow.
For example, California has guidelines for inoperable cars. California car insurance law requires motor vehicles to have current registration if they are driven, towed, stored, parked on public roads or highways or parked in an off-street public parking facility at any time during the registration period.
If you have an inoperable vehicle in CA and it will not be in any of the places listed above, you can apply for non-operational status on the vehicle and pay a planned non-operation (PNO) fee. Once this status is placed on a vehicle's record, it remains until you decide to operate the vehicle and pay full registration renewal fees. When your car is registered as a PNO in California it does not need auto insurance coverage since it is inoperable and off the road.
Check with your state's Department of Motor Vehicles to find out if you can park an uninsured car on the street. You can ask about registration as well since likely if you take insurance off of your car you will likely need to turn in your registration and plates if you do not have car insurance on your vehicle.