New embedded systems are being built with new types of energy sources, including solar panels and energy scavenging devices, in order to maximize their utility when battery or A/C power is unavailable. The large dynamic range of these unsteady energy sources is giving rise to a new class of power-aware systems. They are similar to low-power systems when energy is scarce; but when energy is abundant, they must be able to deliver high performance and fully exploit the available power. To achieve the wide dynamic range of power/performance trade-offs, we propose a new task motion technique, which tunes the system-level parallelism to the power/timing constraints as an effective way to optimize power utility. Results on real-life examples show an energy reduction of 24% with a 49% speedup over best previous results on the entire system.