The most likely cause for malfunctioning tail lights is a blown fuse, followed by a problem with the wiring or the control switch. Other causes include bad light bulbs, faulty bulb socket, or a bad light sensor. The taillights have separate controls, wiring, and sockets for each of the functionalities.
So, if your brake light fuse always blows out, this means that one of the electrical components on the same circuit has issues or problems that cause it to draw out more current than it should. When that happens, one of the components is probably causing the entire circuit to short out, which would then force the fuse to blow.
Front and rear, left and right parking lights are all on separate circuits out of the BCM, so maybe it is a problem with internal BCM control logic for that side. I'd try the old 'disconnect the battery ground cable at the jump-start lug for 10 minutes - master logic module reset trick' to see if that maybe sets things right.
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