How can I turn off the rear parking lights - 2011 Cadillac SRX

Replace that bulb. All those functions are from the same one filament inside those two filament bulbs. One filament is for park lights, and the other is for turns and brake light.

Multifunction switch in the steering column has gone bad

No relay for the parking lights .
When the headlamp switch is moved to the PARK or ON position, battery voltage is supplied from the 15A Park Lamp fuse to circuit 9. Current flows through circuit 9 to the instrument panel fuse block (IPFB). Battery power passes through the IPFB to both license lamps, both front and rear tail lamps, both front park/turn lamps, and all marker lamps through circuit 9. The front marker and front park/turn lamps are grounded by circuit 150 through the forward lamp ground splice pack. The rear tail lamps and license lamps are grounded by circuit 550 at the rear ground splice pack and the right rear body ground.

What year is it ? Parking light or Parking LIGHTS

It's probably the park lamp relay in the rear fuse box , contacts stuck together ! The body control module turns on the relay after receiving a input from the headlamp switch !
The rear fuse box is located beneath the left rear seat !
30
PARK LP Relay
--
F PARK Fuse, LR PARK Fuse, RR PARK Fuse, TR PARK Fuse

There are 2 fuses for parking lights left and right side. so it can be 2 relays. located in the box in the front.

faulty connection ,or check all your fuses from box,check fog lights fuse 1st,

have you started with the basics check the fuses the head lights and the parking lights are different fuses

Park, Tail, Marker Lamps, and License
The park, tail and marker lamps, including the license lamps, turn on when the headlamp switch is placed in either the PARK or the HEAD position, or anytime the headlamps are requested. The RT I/P #1 fuse in the RH I/P fuse block supplies battery positive voltage to both the park lamp relay switch contacts and to the park lamp relay coil circuit. The park lamp relay coil is controlled by the body control module (BCM). When the headlamp switch is placed in the PARK position, the BCM energizes the park lamp relay coil control circuit.
Current flows from the park lamp relay switch to both front and rear park lamp fuses and to their respective park/tail lamps. The front park lamps are grounded at G101. The rear park lamps are grounded at G302. The license plate lamps are a part of the rear park lamp circuit and are grounded at G302. The inadvertent power control feature prevents the park lamps from remaining on more than 10 minutes if left on. If the park lamps are turned on after the ignition switch has been turned OFF, or is in the ACCY position, they will remain on until turned off or until the battery runs dead.

Did you the REAR PARK LP fuse 15 amp in the right insturment panel fuse box.

Check for power to your brakelight switch this located at the top of the brake pedal. I see alot of these go bad.

brown wire is your park LT yellow Left-turn green right-turn. Now the break light my be on it own wire it is red black is ground. but if you have a test light see if the brown wire turns on the test LT when you turn on the park LT or what wire light up when you step on the break pedal. and so on.