This paper analyzes the possible contribution of actuation redundancy in obtaining very high acceleration with parallel robot manipulators. This study is based on redundant and nonredundant Delta/Par4-like manipulators, which are frequently used for pick-and-place applications, and addresses the cases of translational manipulators (also called 3T manipulators) and manipulators with Schoenflies motions (also called 3T1R manipulators). A dynamic model, valid for both redundant and nonredundant manipulators, is used to analyze the moving platform's acceleration capabilities: (i) at zero speed and in any direction and (ii) at zero speed in the "best" direction. The results show that actuation redundancy makes it possible to homogenize dynamic capabilities throughout the workspace and to increase the moving platform's accelerations. Designs of redundant Delta/Par4-like manipulators capable of high acceleration pick-and-place trajectories are presented for both 3T and 3T1R manipulators.