An integrated design method is proposed to integrate underactuated gripper into a 3-[R(RR-RRR)SR] kinematic redundant (6 + 3)-degree-of-freedom (DOF) parallel robot for grasping objects with surface curvature. The underactuated gripper is composed of three identical fingers, three identical parallelogram mechanisms and three identical bevel gear pairs, and is symmetrical in rotation at 120°. Each finger can be individually actuated by each redundant DOF of the (6 + 3)-DOF parallel robot. The platform and the gripper can be operated using only the nine actuators located near the base, without requiring additional ones for the gripper. The kinematics of a global robot composed of the (6 + 3)-DOF parallel robot and the gripper is modeled, and its inverse kinematics and forward kinematics models are derived. Grasping types and grasping stability between the proposed three-finger gripper and existing two-finger gripper are compared and discussed. The underactuated three-finger gripper is better equipped to overcome the arbitrary wrench exerted by the grasped objects, which can stably grasp more types of objects than the existing two-finger gripper. Finally, the kinematic simulation is carried out, and the basic dimensional conditions of the target object that can be grasped are presented.