Rear running lights /rear taillights not working?

Hello, it looks like there is only 1 ground wire for both rear exterior light assemblies, although there are a couple different designs with these. Looking at aftermarket vs OEM diagrams, there are some differences. Do you have 4 wire harness connectors going to each rear lamp assembly? One wire being Black (Ground)? Or are there 5 wires? I'm going to post this aftermarket diagram first just to see which setup you're working on.
This is the only aftermarket diagram for the exterior lamps, the OEM differs a bit. The purple wire coming in from page 2 is a 12volt feed that's for both sides coming from the relays and fuses. This one shows a logic chip in the rear assemblies for the LEDs, I will post page 2 as well. I assume there is only one output from the BCM for the running lights but that ground G402 is supposed to be for both.
Diagrams 2 is labelled Park Lamps without T90,
diagram 3 is with T90 option.
Of course, the OEM show each side using a different Ground G401 and G402
Are you able to scan the BCM for any codes?
Here are the locations for Grounds G401 and G402, with both sides being out though this is where aftermarket diagrams might be incorrect.

This is some additional information on the rear lamp's operation, it appears that the Brake Lamps, turn signals, Parking lamps, License plate lamps, and the Reverse Lamps all use the same grounding locations. So, I don't think this would be a grounding issue. Service info also stating there are Left and Right Tail lamp Outage detection circuits, which would be the purple wire for the Right (Pass) side and the yellow wire on the left (driver) side. I've added the connector pinout for both (with T90 and without T90) option for the rear tail assembly. So, if you do have this brown/white wire for the right side and the purple wire for the Left side, it looks like that is where power is supposed to be coming from for the Parking (Tail) Lamps, obviously service info here is incorrect. The last 2 diagrams are supposed to be OEM though, where Pin F should be the feed for the Park lamps from their either separate relays (T90) or from one relay (-T90). Relay LT/POS/LP Relay to each fuse (LT/POS/LP Fuse 10A) - (RT/POS/LP Fuse 10A).
If there is no power on the purple wire (driver) and Brown/White wire (passenger) then it has to be an open circuit to the rear assemblies. Hopefully some of this info is helpful, it looks like a mess. Sorry about that.
I'm posting a little info on a different case study that may or may not apply. But it appears the STX also seems to use those same wire colors for the front park lamps. So, it may be an operation that is confusing here. Check that out and see what you think. If the Park lamps come on at different shift positions.

One section does list the Tail Lamp relay being in the rear fuse block, so passenger side rear of luggage compartment, that is in the rear of the vehicle, and I'm not at all a 100% sure on these wiring diagrams, This is the diagram of that, showing the Purple wire and the Brown/White wire that run to the rear tail lamps, in this it appears that they run the wires for the front marker lamps from that rear fuse block, but that doesn't make much sense, So I would check this rear fuse block, it's probably behind some paneling in the trunk area. The fuse labels don't make much sense either, the connector diagrams definitely have errors in them. You can test the ground wire at the rear lamp assembly by hooking it into a power wire and touching the black ground wire and see if your meter reads 12-volts if that's what you are using. LEDs don't take nearly the amperage of incandescent lamps and that's why they show resistors in-series in the circuit with the LEDs.

Each version whether it's the T90 option or not shows an 8 pin Tail Lamp connector, some pins are not used in the non T90 design but according to the OEM diagrams Pin F is used in both cases for the rear Tail/Park lamps, but you can see in the 4th diagram they have the left side marked incorrectly. In either case you should have a brown/white wire on the Right side with power for the Tail lamps,

It should be the same for the driver side rear, but a purple wire, and that's coming from the Relay for those Tail lamps, so if you have power on one of those wires you can use the Black wire as your ground for the meter and should read 12v. There are some timeouts programmed into the BCM, but it's supposed to be 10mins so if you test with the vehicle running you should be fine, and I would also check it with the switch set on a manual mode as well as any auto setting it has.

This other vehicle was a 2005 Cad CTS, and its Park lamps were only those small outside quarter panel lights. The Front marker lamps only worked when in Drive, Neutral or Reverse and with the Headlights off (5th diagram). So, that might be the case here as well, but we wouldn't know because they don't have anything labelled correctly. I do hope you aren't chasing a normal function here. But it might be the case. It looks like the BCM uses the same LEDs as Turn signals and just pulses them off and on. With the 3 sets of LEDs on each side, 1 is for the Reverse lamps which need to be a white color, then Stop lamps. Whatever lamps turn on with the License plate lamps, if they are the small quarter panel lamps in the rear might be the Marker/Park lamps, whatever they are called. And the ones that come on in Drive might be tied into the daytime running lamps or some sub system.
Sorry it's so confusing but have someone stand in back and tell you what lights come on or off with the headlight switch positions.

Sorry for the confusion with the wiring diagrams, there are just a lot of variables in the design with these vehicles. I had seen that the other year CTS had some strange function when it came to running lights, and a DRL sensor involved. I'm just going through your last few posts here to see where you're at right now, I agree it's not a bulb issue, since it looks like each side uses a different ground. But it was looking like some of the OEM diagrams might not have been correct. That does happen, where the OEM is incorrect but the aftermarket and then the other way around as well. But it does help to see the actual connector you're dealing with now, I am counting 5 wires per connector, is that correct?
A lt green, dark green, purple, tan/white, and a black?
The only wire I see that has a white stripe is the pass side F pin, which is brown/white according to the diagrams, and its coming from the Rear Fuse Block LT/POS/ LP Relay then out through the RT/POS/LP Fuse 10amp. Which is supposed to also power the Right Front Park and Marker Lamp, as well as the License Plate lamps.

To check for a short to ground, I use a test light hooked to battery positive, and I make sure that the circuit I'm testing does not go through any modules so I'm not pulling too much current through that module. Or I would use an LED test light which draws very low current.
You could unplug the license plate connector and take the bulbs out of the circuit and see if the circuit lights the test light, showing a path to ground before the license plate lamp bulbs (the load) for the circuit.
I use that method so when I find the short, like moving a wire away from the short, my test light will go out.

These are all the OEM diagrams for all the rear lamps, I labelled them exactly how they are listed, you can see why the confusion. Besides each of these listing a Logic board internal to the rear lamp assemblies.
Are the front marker lamps coming on when you're powering up the rear ones from an aux power source?