Do you have a multi meter if needed to check Volts and OHMS?
If the head light switch is ok, the problem should be one of three things, a short somewhere in the wiring harness, contacts in the relay shorted (burned together) or the rocker switch is not opening. Those rocker switches rarely go bad. When most headlight switches go bad they normally will not turn the lights on.
Power is fed to the Tail Illumination Relay and the rocker switch behind the steering wheel. (the two are basically connected in parallel) When the relay closes (using the headlight switch) or the rocker switch is flipped the parking lights go on.
Try this first.
The Tail Illumination Relay is located at the top center of the under dash fuse panel. There should be 3 relays there side by side.
Remove the center relay, connect the battery and see if the lights are off. If the lights are off, then replace that relay with one of the relays right beside it. Did the lights stay off or come back on?