Redundancy has been exploited in robotics primarily to avoid mechanical interference between the robot and its environment, and to avoid kinematic singularities. Separately, variable-stiffness mechanisms have been used to mechanically tailor the stiffness properties of robots to enhance functionality in certain types of robotic interactions in the absence of sophisticated control schemes. In this paper, we propose to combine the principles of these two approaches into a single device with potential application in upper-limb rehabilitation, where the ability to control the stiffness of the interaction felt between the human user and the robot’s end effector is desirable, and safety concerns caused by kinematic singularities are also of high importance. We present a redundant planar parallel robot in which the redundant links/joints also serve as variable-stiffness mechanisms. We lay out the kinematic model, the stiffness model, and their interdependencies, and illustrate the resulting robot behavior through simulations.