You have to test the multifunction switch and make sure the wire harness is good. It most likely the multifunction switch.
According to the owner's manual for a 2002 Ford Explorer Sport (2 door) in the fuse panel at the end of the dash (visible with driver's door open) in location # 13 there is a 15 amp fuse for stoplamps, high mounted stoplamp, and speed control. With the third brakelight working , the fuse must be good. Have you checked the 2 bulbs for the taillights/lower brakelights to see if 1 of the filaments is broken in each of the 2 bulbs ?
The brake light switch is located near the top of the brake pedal. When the pedal is depressed the switch is activated and sends current to the brake lights. To access the switch, the under dash panel needs to be removed.
If all tail lights/stop lights are out could be fuse If only one side is out --probably bad fixture on that side
check brake light switch
Check the fuse, bulbs/assy., and wiring another thing to check is where the bulbs connect. sometimes like my 94 one of the connectors in the bulb housing was broken. also there is a brake light sensor on the brake petal that is turned on when you depress the brake. usually near the top of the brake petal
Check the bulbs - they are double filamented- smaller for stop and larger for tailights
I am not sure on a ford but i had about the same problem on a GM car i had and it was the turn signal switch i replaced it and all the brake lights worked fine you might try that
Depending on the year, there may be one fuse for all three brake lights, or two fuses, one for the highmount brake light and one fuse for the two lower brake lights/tail lights. So, if all three brake lights are out, it could be just one fuse or perhaps the brake light switch which is on the brake pedal metal arm assembly.
Where is brake light fuse for 2004 Ford Explorer located
Probably a faulty brake light switch or the brake light switch is installed incorrectly.
its gotta be a ford a problem with the center brake light circuit causes the lights to go out i believe it is a diode
From the Ford Dealer
The left brake light works the right brake light doesn't and the bulbs is good
Check for a brake light fuse
Brake light switch is broken or needs adjustment.
Check your brake light switch at hte brake pedal.
The brake lights won't turn off because the brake light switch on the pedestal of the brake pedal needs to be replaced.