Two novel parallel mechanisms (PMs) employing two or three PaPaRR subchains are suggested. Each of those two PMs has translational 3-DOF motion and employs only revolute joints such that they could be adequate for haptic devices requiring minimal frictions. The position analyses of those two PMs are conducted. The mobility analysis, the kinematic modeling, and singularity analysis of each of two PMs are performed employing the screw theory. Then through optimal kinematic design, each of two PMs has excellent kinematic characteristics as well as useful workspace size adequate for haptic applications. In particular, by applying an additional redundantly actuated joint to the 2-PaPaRR type PM which has a closed-form position solution, it is shown that all of its parallel singularities within reachable workspace are completely removed and that its kinematic characteristics are improved.