Is it possible to use a car's onboard diagnostic port to toggle the headlights, interior lights, or parking lights? If not through the OBD-II port, how would I go about remotely toggling the lights on a car?
The short answer is: no - not without the manufacturer's diagnostic tools.
The long answer is: maybe. Modern cars have most (if not all) exterior and interior lights controlled by the body control module (BCM). Most (if not all) switches are connected to the BCM as well. This provides a lot of flexibility. A prime example is if you leave your headlights or interior lights on, the BCM can override and turn them off after a timeout to prevent a dead battery.
Another interesting example is the way turn signals operate. In older cars, switching the turn signal on would close a circuit from the battery to a component with a spring in the path. This spring would expand from the heat as current passed through it. Once it expanded enough, it would open the circuit, and as it cooled down, it would shrink and close the circuit again. This in operation generated a clicking sound that we're all too familiar with. Software in the BCM of modern vehicle's control the turn signals, and the "clicking" sound is simulated (usually by a transducer in the instrument cluster).
It is possible to control (or actuate) the lights via the OBDII port by issuing an "input/output control" diagnostic command (as defined in ISO-14229 Unified Diagnostic Services) with the ID of the light you want to control. The problem is finding this ID, as they are vehicle manufacturer specific. Only the official diagnostic software/hardware from the manufacturer contains this information.
It would be possible to brute force the IDs by sending the IO control command for every possible ID in the system supplier specific range (which is around 500 IDs). This would take a lot of knowledge, time, and effort - but is not impossible.