This paper introduces a novel family of singularity-free kinematically redundant planar parallel mechanisms that have unlimited rotational capabilities. The proposed mechanisms are akin to conventional three-degree-of-freedom planar parallel mechanisms. By introducing a novel kinematically redundant arrangement, four-degree-of-freedom parallel mechanisms are obtained that can completely alleviate singularities and provide unlimited rotational capabilities. The kinematics of the mechanisms are derived, and the Jacobian matrices are obtained. It is shown that the singularities of this type of mechanism are governed by the orientation of a passive link connecting the redundant leg to the platform and that the latter orientation is easily controlled using the kinematic redundancy, thereby alleviating all direct kinematic singularities. An example mechanism is proposed, and a prototype is demonstrated. Example trajectories that include full cycle rotations are shown. The prototype also illustrates the use of the kinematic redundancy for an auxiliary task, namely grasping.