Third brake light? I'm not sure what you mean as these trucks only have two brake lights.
But on the backup lights the problem is frequently either the switch in the transmission or the wiring back by the lights. So I'd start by looking to see if the wiring is connected to the switch on the transmission. Then I'd follow it on back to the lights themselves.
Gary, AKA "Gary fellow":
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Dad's: '81 F150 Ranger XLT 4x4: Down for restomod: Full-roller "stroked 351M" w/Trick Flow heads & intake, EEC-V SEFI/E4OD/3.50 gears w/Kevlar clutches
Blue: 2015 F150 Platinum 4x4 SuperCrew wearing Blue Jeans & sporting a 3.5L EB & Max Tow
Big Blue: 1985 F250HD 4x4: 460/ZF5/3.55's, D60 w/Ox locker & 10.25 Sterling/Trutrac, Blue Top & Borgeson, & EEC-V MAF/SEFI