Press the Brake the Parking Lights come on.

It could be. You see the hazard switch complicates the TS switch, and sometimes they sort of fall apart internally. Basically the hazard switch interconnects all four TS wires, front rear, left, right and also hooks them to the hazard flasher output

Here's one thing you might do..........Pull loose the TS connector at the column look up the colors for the 4 TS lamp wires, and check continuity at the switch side of the connector. . ONLY the two rear TS wires should show continuity with each other. with the switch centered. If you get continuity from those two to the two in front............it's the switch

I would not wiggle it too much in trying to test. You could "clear" the fault, and then not be sure what to look for.