The microgripper plays an important role in the microassembly system, which directly contacts the target object. The working range of a normally open or normally closed gripper is limited. In this paper, a monolithic flexible microgripper with dual piezoelectric-actuation modes is presented, which combines the characteristics of normally open and normally closed grippers. It has the advantage of enhancing the adaptability of the gripper for grasping objects of different sizes. Different types of bridge mechanisms and parallelogram mechanisms are used in the displacement amplification structure of the gripper. This design can improve the magnification efficiency and linearize the tip motion. According to the kinematics model, the displacement magnification of one-side gripper is analyzed. The crucial structural parameters affecting the magnification were selected to complete the structural optimization of the clamp. Finally, the optimized gripper is simulated and verified by performing finite element analysis. The simulation results reveal that the designed gripper has sufficient clamping stroke and good dynamic performance.