The bulb failure relay compares the current draw between one side of the car vs the draw on the other side of the car. If it sees a mismatch in amperage it illuminates the warning light. So when you power BOTH tail lights off of the same side you are exaggerating that mismatch and the warning light illuminates.

Sometimes that sensor is so sensitive that you have to make sure you have not only the same wattage bulbs on both side, but the same BRAND of bulbs.