1996 Silverado. All 3 brake lights don't work. Flashers do.

Check the bulbs and grounds at the trailers marker lights.

Troubleshooting No Running Lights on Trailer but Turn Signals and Brake...
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Thread Running lights on trailer don work but turn signals and brakes do

replace your flasher, it controls both turn signal, flashers also brake circuit runs througth.

turn on the emergency flashers(e-flashers)if they work check the bulbs at the same time.if this is good then check the fuses.you may have a problem with the turn signal flasher needing replacing.this is a different flasher than that of the e-flasher,looks the same but a different flasher.usually located on or around the fuse panel under the dash.the brake light problem could be a misadjusted switch or a bad switch.the switch is at the base of the brake pedal and has two wires to it.one wire will have power all the time and the other wire will only have power on it when the brake pedal is pushed.if not then the switch will need replacing.check for broken wires at the switch as these sometimes break causing a problem.

the bulbs are 2 filaments right? there has to be 2 wires unless there is a ground wire (3 wires) every trailer has to have a Running light, Left turn, right turn, and ground. th ebrake and turn signals work the same wires. you dont say what kind of truck you have, but some trucks have fuses for trailer lighting. put the 4 way flashers on and see whats wrong. if you have flashers you should have brake.

Have you checked the bulbs? Some people don't beleive all the lights can burn out at once but when one brake light is on they are all on and for the same amount of time.
If the bulbs are fine then check the connection where the trailer plug is to the main harness. Some of those connectors go between the main harness and the tail lights and the connector can go bad.

check brake switch by the back of the brake pedal and make sure it is working

See if the turn signals work,and the flashers,the problem,is most likely in one of the flashers,they are stuck,the flasher.

If you have an aftermarket trailer towing plug on it, that's where i would look first. Something is shorting out in there, or you have a broken wire or ground somewhere. Proabably near the back where gravel stones can hit.

Check at the light socket in the rear, where your lights don't work. You need a test light or a voltage meter, and check to see that you have a groun there, power for your parking light, and power to the brake light. You should have power for the parking lights seeing that it works, and a ground. Just verify that your power for the brakes is getting to the bulb, or not. Just a good place to start.
And you are correct that if the third brake light works (top) your brake light switch is fine.

Possibly blew out the hazard flasher unit. Check the flasher unit (usually there are 2, one for directionals and one for flashers).