The use of continuous and flexible structures instead of rigid links and discrete joints is a growing field of robotics research. Recent work focuses on the inclusion of continuous segments in parallel robots to benefit from their structural advantages, such as a high dexterity and compliance. While some applications and designs of these novel parallel continuum robots have been presented, the field remains largely unexplored. Furthermore, an exact quantification of the kinematic advantages and disadvantages when using continuous structures in parallel robots is yet to be performed. In this paper, planar parallel robot designs using tendon actuated continuum robots instead of rigid links and discrete joints are proposed. Using the well-known 3-RRR manipulator as a reference design, two parallel continuum robots are derived. Inverse and differential kinematics of these designs are modeled using constant curvature assumptions, which can be adapted for other actuation mechanisms than tendons. Their kinematic performances are compared to the conventional parallel robot counterpart. On the basis of this comparison, the advantages and disadvantages of using continuous structures in parallel robots are quantified and analyzed. Results show that parallel continuum robots can be kinematic equivalent and exhibit similar kinematic performances in comparison to conventional parallel robots depending on the chosen design.