I've never seen a car where the "first click" of lights (what I presume you're calling parking lights) doesn't turn on the taillights. In most cars, they're even on the same fuse.

(Yes, there are cars with true one-sided parking lamps that will only illuminate one side front-and-rear to mark the edge of the car parked closest to the road.)

I suspect what you're actually seeing is drivers using only the daytime running lights. Those do not turn the rear lamps on.