When the brakes are applied, not only does the driver's side light not come on, but the parking lights go off. What would cause this?

NOT ABS lamp , right?
the parking brake is adjusted first pre the service manual then
the switch must be adjusted to for closure brake released;
i use volt meter that switch after install and make sure it works
if not RTM read the service manual , on topic

this care is ODD
see this chapter in the service manual if opted day1.

Electronic Parking Brake Description tip if ice berg

Vehicles with the electric parking brake have a switch in the center console, which takes the place of the manual parking brake system, the foot pedal and release handle. In case of insufficient electrical power, the electric parking brake cannot be applied or released.
Occasionally, the electronic parking brake will automatically apply when the vehicle is not moving. The electronic parking brake performs a operational check every 50 key cycles. When this check is being performed, the park brake light on the instrument cluster becomes illuminated. This function does not affect the normal operation of the vehicle.
Electric Parking Brake Control Module The parking brake control module has an internal motor, apply actuator, release actuator, and temperature sensor. The parking brake control module also contains the logic for applying and releasing the parking brake when commanded by the park brake control switch. When the parking brake control module receives a signal from the switch the internal circuit board temperature is checked to verify it is within operating range before the control module performs the requested operation.
The electric parking brake module receives hard wired signals from the apply/release switch and communication enable. The electric parking brake module receives wheel speed sensor signals, engine torque, gear information, brake pedal signal, and etc. through GM-LAN.

AND MORE, LOTS MORE, READ THE MANUAL
PAY LIKE i MUST.

https://eautorepair.net/Marketing/Default.asp

WHEN YOU REPLACED THE BULB IT CAUCED A GROUND SHORT. LOOK FOR BAD GROUND WIRE, THAT IS BACKFEEDING THROUGH WIRE HARNESS.

It sounds as though you have either a frayed wire between the brake control circuit and tail/running light circuit of the truck side of the plug, or someone wired the truck side of the plug incorrectly.

Think about it. The parking brake lights come on when the controller is activated. More than likely the tail lights are also.

The tail lights are also going to come on when the headlights are on. This would cause the trailer brakes to be activated IF there is a short in the TRUCK side of the plug itself.

That's where I would personally start investigating, The truck side of the trailer plug.

Hope this helps.

Kia Sorento parking brake or brake fluid dash warning light - this light is illuminated when the park brake is applied, the warning light should go out when the parking brake is released. If the light does not go out it indicates that the brake fluid level in the reservoir is low and you need to pull over and top up at your first opportunity.

1-check parking brakes not applied
2-check brake fluid
3-if it is too low have the brakes checked they might be wearing out it doesn't mean you have a leak but when the brakes wear out the become thin causing the caliber piston to push out more lowering the oil level in the reservoir

I assume you have replaced the lamps/globes with new ones, if not this is the first thing to try.
If your car is fitted with a "faulty light" sensor, ( i.e. if a warning light on the dashboard comes on when you apply the brake and a brake light bulb is not working), this may be the problem. Follow the wiring for the rear lights from the light clusters inside the trunk and you may find the wiring harness plugged into an encapsulated unit in the rear somewhere. Test if power is coming out of this unit when brakes are applied. Or, remove the covers on the brake light clusters from inside the car and check the wiring to the rear window brake light. Very often this is connected to the same ground as the drivers side light. However, the parking and tail and reversing lights usually work off the same ground, so this seems unlikely.

Apply the parking brake with the vehicle parked, engine turned off and transmission in Park. Release the parking brake completely, then apply it again, and ensure that it is fully released. Turn the ignition on, check to see if the Parking brake warning light is on. If it is NOT on, apply the parking brake and check to see if the warning light comes on. If the light come ON wihen the Parking brake is applied,.....fully release the parking brake and check if the warning light goes OFF. If it does NOT go off, then an adjustment is needed at the parking brake switch.

Hello,

If the light is on all the time, it could mean you simply forgot to release the
parking (emergency) brake. The brake warning light remains on when the brake is set
as a reminder. If the brake warning light remains on when the parking brake is
released, then something is amiss.

In some cases, the brake warning light may remain on if the switch on the parking
brake pedal or lever is misadjusted. Nothing is wrong with the brakes and a simple
adjustment should fix the problem.

If the brake warning light comes on when you apply the brakes,
or is on continuously, it means that hydraulic pressure has been lost in one side
of the brake system or that the fluid level in the master cylinder is dangerously
low (due to a leak somewhere in the brake system).
In either case, the fluid level in the master cylinder should be checked.
Adding brake fluid to the master cylinder reservoir may temporarily solve the
problem. But if there's a leak, the new fluid will soon be lost and the warning
light will come back on.

Brake fluid leaks are serious because they may cause the brakes to fail!
So avoid driving the vehicle until the system can be inspected to determine what's
wrong. Leaks can occur in brake hoses, brake lines, disc brake calipers,
drum brake wheel cylinders or the master cylinder itself.
Wet spots at hose or line connections would indicate a leak that needs to be fixed.

Leaking brake fluid can also contaminate the brake linings,
causing them to slip or grab. The uneven braking action that results may cause the
vehicle to veer to one side when the brakes are applied. Brake shoes or pads that
have been contaminated with brake fluid cannot be dried out and must be replaced.

Hope this helps.

lopen your tail light assy... check/replace bulbs common problems bulbs filament grounded if you have a tester, test brake light and reverse bulbs on driver side only.... if not check for lose groundline on wire of reverse light... if you have a lose ground on reverse line, the reverse light only lits up when braking becoz it shares the ground line on brake...

System has a fault and needs troubleshooting.

Test the connectors that the lights go in with someone applying the brake. These connectors are not very well made. You may need a new connector on the side that's not working.