Minimally invasive surgery (MIS) is one of the most challenging techniques for robot designers due to the limited size of access points, the high miniaturization level, and the dexterity needed for performing surgical tasks. Conversely, only a few microfabrication technologies are currently available for developing such small-sized systems, which allow safe operations in human bodies. In order to match these challenges in MIS, both design and integration of actuation systems should proceed in parallel with an identification of most effective transmission mechanisms and kinematics. In this paper, an origami parallel module that generates two rotations and one translation is integrated with a twisting module and a compliant gripper to form a novel four-degree-offreedom grasper. The rotational motion leads to the pitch and yaw motion of the gripper, while the translational motion is converted to a roll motion of the gripper via the twisting module that is stacked on top of the parallel module. In light of plane-symmetric properties of the origami structure in the parallel module, both inverse and forward kinematics are resolved with a geometric approach, revealing a unique joint space and a kinematic mapping of the parallel module, leading to the design of two sets of on-board actuation systems. During the analysis, bending motion of a central spring and static properties of the compliant gripper are modeled using finite-element methods. The structure of the twisting module for motion transmission of the grasper is designed and fabricated using origami folding techniques. Gripping forces of the compliant gripper are evaluated in experimental tests. Further analyses of the system performance are addressed in accordance with the scaling ratio of miniaturization and the scalability of the system is demonstrated by a millimeter-sized origami parallel module produced by the smart composite microstructure fabrication process.