Since the brake lights would pose a large current draw, it may be a weak ground.
Check the ground near the battery. Almost directly below the positive terminal.
Also, check the ground connections between the block and chassis. That should
be the ground strap on the battery side near the hood hinge.
Check to see if the same problem occurs when you turn the headlights on.
It may be that the regulator in the alternator is weak and slow to respond to
the additional amp draw.
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