It depends on the size of the car park and the inclinations of the individual mapper.

If a car park serving a block of flats or commercial building is clearly visible on the aerial imagery, my inclination would be to map it and tag access appropriately.

Most people wouldn't usually map private parking in a paved over front garden.

If there's a gate, you can add a node on the driveway and tag with barrier=gate + access=private + locked=*
https://wiki.openstreetmap.org/wiki/Tag:barrier%3Dgate