blinker indicator in dash

The TS switch "should" have absolutely no connection to the parking circuit. And no, "all of the lights" do not go through there. Only ones that do are front and rear turn. TS lamps in rear do same job as brake so they in reality are 'not' separate.

Most logical might be something like a cross in the harness or bad TS bulb, or ground problem at cluster. First thing "I'd try" is disconnect the rear harness connector in the kick panel and see if it goes away