Robot manipulators are often deployed in tool-symmetric tasks, which only requires defining end effector position and pointing direction. In this case six-axis serial industrial robots and full-mobility (spatial) parallel robots have one degree of task redundancy. Using Cardan angles as orientation coordinates, a unified formulation of the position-level and second-order inverse kinematics problem is set up for both robot types. An efficient scheme for difference-quotient approximation of gradients of performance criteria for projection into the task redundancy's nullspace is presented. The simulation example of a hexapod robot shows that avoiding and exiting parallel robot singularities of type II is possible with the nullspace of all joints. The nullspace controller scheme can be used in offline trajectory optimization and in online motion generation.