We introduce an origami-reinforced parallel continuum robot which is capable of maintaining the orientation of the end effector regardless of the bending shape. The cross-routing tendons provide an effective actuation of the robot because the constant length of the backbones prevent the actuation of parallel arrangement of tendons. We utilise the arclength relationship of parallel curves to show that parallel backbones of equal lengths and constant spacing between the backbones enable the constant orientation of the end effector. The origami shell is introduced to increase the torsional stiffness of the continuum robot and minimise any twisting. This leads to planar and parallel bending of all backbones. Planar quintic Pythagorean Hodograph curves are utilised for shape reconstruction, which is more accurate as compared to the curves with piecewise constant curvatures. The concept of this continuum robot and the accuracy of the reconstruction are validated experimentally.