This paper introduces a novel design for a robotic hand based on parallel mechanisms. The proposed hand uses a triple-symmetric Bricard linkage as its reconfigurable palm, enhancing adaptability to objects of varying shapes and sizes. Through topological and dimensional synthesis, the mechanism achieves a well-balanced degree of freedom and link configuration suitable for reconfigurable palm motion, balancing dexterity, stability, and load capacity. Furthermore, kinematic analysis is performed using screw theory and closed-loop constraints, and performance is evaluated based on workspace, stiffness, and motion/force transmission efficiency. Finally, a prototype is developed and tested through a series of grasping experiments, demonstrating the ability to perform stable and efficient manipulation across a wide range of objects. The results validate the effectiveness of the design in improving grasping versatility and operational precision, offering a promising solution for advanced robotic manipulation tasks.