2006 CTS right and left rear stop lights do not work. Bulbs are OK. Center high mount stop light works. Rest of rear tail light assembly works. Only the right and left stop lights not working.

Light bulbs require power and ground in order to work. The ground circuit for the left rear combination lamp is shared between the turn lamp, stop lamp and tail lamp. Because the tail light still works in that specific combination lamp, we know that the shared/common ground for that assembly is OK. The power circuit for both stop lamps (left and right) have the same origin; voltage flows from the STOP LP FUSE (15A) through the DBC RELAY (when the stop lamp switch is pressed) and then to each stop lamp via their respective circuit branches. Because the right stop lamp works, we know that the shared/common fuse, relay and switch are all OK. I'd suspect an open in the left stop lamp power supply branch and would recommending checking for voltage on the green & black wire at the left rear combination lamp assembly to confirm. Regarding the left lamp on the hatch, the wiring diagram indicates these are not stop lamps rather tail lamps only. I'd recommend installing a tail lamp bulb instead of a brake light bulb into that assembly as a first step. You can read more here: https://www.yourmechanic.com/article/how-to-replace-your-car-fuse-box-by-jessica-howe

What's a DHS ?
Stop Lamps

The STOP LP fuse in the rear fuse block supplies battery positive voltage to the normally open stop lamp switch. When the brake pedal is pressed, the stop lamp switch contacts close and battery positive voltage is supplied to the center high mounted stop lamp (CHMSL) and the turn signal/hazard flasher module stop lamp switch signal input. In response to this input the turn signal/hazard flasher module supplies battery positive voltage to the left and right LED tail lamp assemblies. The CHMSL and left tail lamp have ground at G402. The right tail lamp has ground at G401. The stop lamps illuminate when they receive both power and ground.
es battery positive voltage to the left and right LED tail lamp assemblies. The CHMSL and left tail lamp have ground at G402. The right tail lamp has ground at G401. The stop lamps illuminate when they receive both power and ground.

Kevin , this is a very complex way of stoplamp control , I will post GM service Description and Operation of how they work an let you deside if you want to tackle this ! Do the center high mount brake lights work ?
The brake pedal position sensor is a resistive device used to sense the action of the driver's application of the brake pedal. The brake pedal position sensor provides an analog signal that will increase as the brake pedal is pressed.
The brake pedal position sensor receives both a 5-volt reference voltage and a low reference signal from the electronic brake control module (EBCM). When the brake pedal is pressed, a variable signal is applied from the brake pedal position sensor to the EBCM through the brake sensor signal circuit. The fuse block left rear supplies battery positive voltage to the bass relay switch through the BASS fuse. One side of the bass relay coil is permanently grounded at G402. The EBCM supplies voltage to the bass relay coil upon receiving the signal from the brake pedal position sensor. The switch contacts of the bas relay close and battery positive voltage is applied through the bas relay switch to the center high mounted stop lamp (CHMSL), and to the turn signal/hazard flasher module through the stop lamp switch signal circuit. The ground for the CHMSL is applied from G402. The turn signal/hazard flasher module then supplies voltage to the left and right stop lamps in the tail lamp assemblies that have ground at G401 and G402. For export vehicles the stop lamps receive battery positive voltage directly from the bass relay.

There could be a code stored in the EBCM - electronic brake control module - DTC B3903 Stop Lamp Relay Circuit
DTC C0278 Brake Pedal Position Sensor Not Calibrated
there are several other codes that pertane to the brake lights . you could try Replaceing the bas relay !
Fuse Block - Left Rear
In the passenger compartment, under the left side of the rear seat at carpet.

does the high mounted brake light work? the brake switch has 2 contacts one for the high mount (LT BLUE/BLACK) and one for the tail brake lights. The tail/brake light wire is LIGHT BLUE. the main connector between the brake light switch and the tail lights is C208 which is located behind the rear seat, left side. So if there is power coming out of the brake switch OK and no power at the tail lights, check that connector. Hope this helps.

Since the high mounted brake light works you know that the brake light switch is working. Now you need to check the brake light bulbs in the rear tail light assemblies. The brake lights are separate from the tail light (parking or running lights). There are two screws that hold the rear light assemblies in place.

Remove the tail lamp housing and test for power at the wiring connector. Does the center high mount stop lamp operate correctly? The lamp circuit boards have seen many problems.

Fuse number 7 (15A, Horn Stop) in the under hood box should be your brake lights. If you are referring to your high mount brake light as the third light, I believe they work on the same fuse.
I would check the bulbs, although it may not be likely that both would go at the same time. There are three bulbs in the unit: brake/taillight, turn signal, and backup.
The units themselves come out pretty easy. Just remove the two phillip's head screws and pull the unit back to get it out. Then turn sockets 1/4 turn and out to check the bulbs. Bulbs pull straight out.
Hope this helps you.

That is a little strange, but could be just bulbs have blown. Switch must be ok if high mount works, so check bulbs first. Unless there is a separate fuse you could check that too.

You may or may not see a burnt out bulb. The dash light can also mean there's a current issue (ie none of the bulbs are lighting)

2 reverse bulbs
2 license plate bulbs
2 turn signal bulbs
2 rear sidemarker bulbs (side of your tail light assembly)
2 running light bulbs which are the same as your stop light bulbs in the tail light assembly
1 to 6 high mount stop light bulbs (if they're in the window and varies on models)
1 strip of LED brake lights if you have a spoiler

make sure u have put the right bulbs in. rear brake lamps are dual filament....