Dexterous hands are an important end-effector of robots, but their relatively low carrying capacity, small workspace and poor task adaptability are the key factors that restrict their wide application. To overcome these shortcomings of dexterous hands, a novel Lie-group-based synthesis method that extends the 3-[P][S] parallel mechanisms (PMs) to dexterous hands is presented, and a class of three-finger dexterous hands with parallel finger structure is obtained. The multimode operation is proposed by designing a double-slider palm that provides the hands with a large workspace and high task adaptability. The operation types are presented, and the dexterous in-hand manipulations in all modes are analyzed by means of Lie group theory. In addition, the equivalent structural characteristics of pinching objects are classified to elucidate the motion types and the rotational properties of the pinched objects. The inverse kinematics of fingers is presented and is used to identify the input–output relationships. Finally, the workspaces of the fingers are determined according to the result of the inverse kinematics, and the relationships between the size and displacements of the pinched object are presented. The proposed dexterous hands overcome the problems of low carrying capability, small workspace, and weak in-hand manipulation ability that are encountered with the traditional dexterous hands, which are underactuated and are built with a series finger structure, and can be potentially applied to various application domains, such as services, industry, and rescue.