Parallel uses the same amount of power as serial. Power is V * I, in serial connection V is the sum of all the LED voltages but the current is the same through them all. In parallel, the voltage is that of one LED but the current is the sum of all the LEDs.
It's a case of high current with low voltage or the other way around, when you multiply the two you get the same result.
However, if you have the voltage available for a serial connection, you need fewer components. As the current is the same throughout the loop, you only need to regulate it once. If you wire in parallel you have to regulate it for every LED. You can't reliably just connect LEDs across each other, the one with lowest forward voltage will divert current away from the remainder and you will get different brightness from them.
Brian.