On my '99 F150 when I turn on the dome light the dash lights and parking lights come on too. Vise-versa when I turn on the running lights the dome light comes on.

 

That's an odd one, I would have to say that there is something wrong internally with your light switch on the dash and it is sending power to the wrong places.

You can probably pick one up off of eBay or from a junkyard for about 20 bucks and swap it out to see if that is the problem.

 

I removed the switch and disconnected the plugs from it and the running lights still come on with the dome light when I open the door.

 

I tried putting in a different light switch and got the same thing!