Yes, cleaning it could solve the problem, but then the rear tail light should also be on.
If, for example, it's only the park light coming on, and it remains on after removing fuse 2 (second from top) it's not the ignition switch but a wiring problem downstream from fuse 2.
If power to the parking light is coming from the ignition switch through the running circuit, then the tail light and instrument cluster back lights should also be on (and other cluster lights with ignition on, engine off) unlikely. In this case turning the handlebar left and right might cause the light to come on and off.
If the power is coming from the parking circuit, then the tail light should also be on but not the back lights (or any other cluster lights) as a battery saving feature.
Unlikely, but if you turn the key off and remove it before the key is vertical, it might be in park mode, then that's normal if the tail light is also on. I'm assuming the tail light is working or the red warning light would remain on.