Sounds like you have a wire wire running to ground or a bulb that should be in a single socket actually in a double socket. I would go around and remove a bulb at a time to see if the behavior changes as you take a bulb out of the circuit. Any chance of a past rewiring that may have put a wire from the parking circuit into the signal circuit?
Clearly there is something going on either in the wiring or in a bulb that is circumventing the way the flasher is supposed to work.
Hate being that guy...hey, don't wanna be the guy blinking down the road, lol