What is it that makes the headlights go down, when closing the headlight switch?

Your finger pushing the switch?
Electricity?
couldn't resist. A little more info on what you're trying to figure out will help.
Are yours not working?

I can't remember If I had him change the switch in this long conversion project, but this switch only pulls out to the first stage, and headlights come on with parking lights. In other words, I can't just have parking lights on.

Could I have a wrong switch?

On my '71, first notch out is parking lights.

2nd notch out is headlights and headlights pop up.

1st notch back in is parking lights with headlights up.

2nd notch back in is no lights and headlights drop down.

You should have 2 notches otherwise it is the wrong switch. Maybe a VW part?

That could explain maybe why they don't want to go down.

How do you know what the want?

Also couldn’t resist!