Current robot hand grippers capable of self-adaptive or coupled grasping often cannot perform linear-parallel pinching at the physical end of the gripper, which is widely used in industrial applications. For this reason, this paper introduces a gripper with hybrid grasping modes— the LPCSA hand. It can achieve three grasping modes: linear-parallel pinching, coupled, and self-adaptive grasping. The design cleverly couples two kinds of Chebyshev linear mechanisms to enable flat movement at the end of the finger. It also utilizes the deformability of the parallelogram to achieve self-adaptive grasping. Furthermore, the gripper uses an idle stroke and a special component to facilitate the switch between the three modes. The linear-parallel pinching function is suitable for pinching objects of different sizes on the desktop. The self-adaptive grasping mode can adapt to objects of various shapes and sizes. The coupled grasping mode enables fast grasping of irregular objects. This paper also analyzes the kinematics and dynamics of the LPCSA hand. Combined with experiments, it demonstrates that the LPCSA hand has a wide range of grasping space and stable performance.