The headlight switch is a multi function switch, it is possible for one part of the switch to work and the other part not work. It is possible, as with most all switches for the switch contacts to arch and weld together.
The path is from the fusible link to the headlight switch through a built in circuit breaker to either the parking light position or the headlamp position.
Look at the headlight switch knob, when pushed all the way in, the knob should rest against the the threaded mounting nut. If there is a gap the knob and rod aren't in all the way. The threaded locking nut should be screwed down tight. The knob and rod should be all the way into the headlight switch, it is possible to install the knob and rod partially and not completely. The lock for the knob and rod is under the switch a finger size hole with a locking plunger(button) in there, push the plunger and make sure the knob and rod are pushed all the way flush with the locking nut. Sometimes you have to turn the knob and rod as you are inserting them into the headlight switch, with the locking plunger (button) depressed to get everything seated all the way in. Sometimes it takes a little force to get the knob and rod all the way in, but this is an iffy proposition, if you push too hard you may damage the dash light dimmer part of the headlight switch.
It is also possible to have a partial meltdown of the headlight switch connector, when this happens no telling which wires are going to contact. The headlight harness connector near the air box is also a prime spot for a connector meltdown.
Could be the switch, the switch knob not all the way in, a partially melted connector, or even melted together wiring someplace.
It also behooves you check the size of the parking light fuse. It is not uncommon for that fuse to blow for various reasons. If the fuse is replaced by a higher (than specified) amperage fuse, bad stuff happens, like harness and connector meltdowns.
Most times when that fuse blows it is a problem in the dimmer part of the headlight switch. The dimmer partially shorts out, generates a lot of heat and bad stuff happens. Often the parking light fuse and the instrument light fuse both blow when there are dimmer issues.