I didn't even look at the link or part number lol. When in doubt always go back to what you did last before the problem.
You needed a dual filament bulb but used a single. I don't have a socket in front of me but the proper bulb p215w bulb has 2 offset contacts on the bottom the p21w bulb has a single one in the center. It's possible the wrong bulb connected both terminals on the car side when the brake lights were on or the other way around. If it senses power in the brake light circuit when the brake switch isn't on it could go haywire. Someone else may have a better answer that makes more sense but that's all I got. Even running led bulbs can make it do odd things. I've seen brake warning lights come on the dash when both brake bulbs are burnt out too.
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View my fuel log 2014 Mirage SE wussie cvt edition. 1.2 automatic: 37.7 mpg (US) ... 16.0 km/L ... 6.2 L/100 km ... 45.3 mpg (Imp)