Drawing inspiration from kirigami, this article first presents the crease pattern of a kirigami model which is kinematically equivalent to a Bennett plano-spherical linkage. In terms of the screw theory, distinct closed-loop motion branches of the linkage are revealed. This article then introduces a new reconfigurable parallel manipulator with three hybrid kinematic limbs. Each limb consists of closed-loop subchain, the Bennett plano-spherical linkage, and a R(RR) serial chain. Constraints provided by the hybrid limb are explored by analyzing constraint screws of serial limbs kinematically equivalent to the hybrid limb in different motion branches. The analysis reveals motion characteristics of the moving platform when the parallel manipulator is in different motion branches. The kinematic model provides a unified mapping between joint inputs and outputs of the reconfigurable manipulator in all three motion branches. This article further presents a new inflatable angular pouch motor and fabricated a prototype using a rectangular tile origami base and adhesive fabric. The pouch motors are then integrated with 3D printed prototypes of the Bennett plano-spherical linkage and the parallel manipulator for the purpose of reconfiguring motion branches.