The answer is both increase. Here's the reason why:
When you connect anything in parallel it creates a new loop into which current can flow. The current can flow either through the two bulbs in series or through the one bulb in parallel. The 12 volts of the battery pushes the electrons through both so current increases in the battery. Power is the product of the current times the voltage, so increased current means increased power supplied by the battery. This means that the battery gets used up faster as it is doing electrical work at a higher rate.