Can you cover your car windows when parked?

Its as illegal as using a car cover or tonneau cover in a parking lot. Which means that it isnt.

Is car considered private property?

The road is a public space, but your car is private property. Lets refer to the definition of public space under Executive Order (EO) No. 26: Providing for the Establishment of Smoke-Free Environments in Public and Enclosed Spaces.

Is it illegal to cover up your windows?

Often law enforcement and first responders can have them darker than the rest of the citizens—for various reasons. And there are some places that will print and wrap your vehicle, except for the noted windows—and yes it is fully legal.