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This work presents and evaluates a novel input parameterization method which improves the tractability of model predictive control (MPC) for high degree of freedom (DoF) robots. Experimental results demonstrate that by parameterizing the input trajectory more than three quarters of the optimization variables used in traditional MPC can be eliminated with practically no effect on system performance. This parameterization also leads to trajectories which are more conservative, producing less overshoot in underdamped systems with modeling error. In this paper we present two MPC solution methods that make use of this parameterization. The first uses a convex solver, and the second makes use of parallel computing on a graphics processing unit (GPU). We show that both approaches drastically reduce solve times for large DoF, long horizon MPC problems allowing solutions at real-time rates. Through simulation and hardware experiments, we show that the parameterized convex solver MPC has faster solve times than traditional MPC for high DoF cases while still achieving similar performance. For the GPU-based MPC solution method, we use an evolutionary algorithm and that we call Evolutionary MPC (EMPC). EMPC is shown to have even faster solve times for high DoF systems. Solve times for EMPC are shown to decrease even further through the use of a more powerful GPU. This suggests that parallelized MPC methods will become even more advantageous with the improvement and prevalence of GPU technology.