Whether you can keep an uninsured and unregistered vehicle on the street or in a parking lot will depend on your state's laws. It may be possible to keep the car parked or stored here until you obtain insurance and register the car or it may be illegal and the car may be towed if found without registration or expired registration.
Typically, the state requires all vehicles to be registered and therefore insured.
Likely it will depend upon if the car is parked in a public or private parking lot. If it is public then it would likely be required to be registered and insured. If it is private, it would probably depend upon what signage there is from the lot's owners and again state laws.
Most states require all motor vehicles to registered and while a car is registered for it to have auto insurance on it. If the car is inoperable you may be able to get special status on it from the DMV, you would have to check to make certain.
For example, California has guidelines for inoperable cars. California 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.