This study has designed an easy-to-wear parallel continuum robot-based hand rehabilitation system that supports and enhances the finger’s flexion, extension, abduction, and adduction movements. The primary novelty of the proposed system lies in its ability to focus therapeutic exercises on a single joint, a feature not commonly found in existing rehabilitation robots. A kinematic model of the system was developed, and to perform both kinematic and dynamic analyses, a multibody model was constructed in the MATLAB Simulink environment. Joint angle control was implemented using a nominal controller, and to account for individual uncertainties in joint dynamics, a neuroadaptive controller was integrated with the nominal controller. This approach aims for the neural network architecture to learn these uncertainties during control iterations and incorporate them into the control, resulting in a robust controller. Thus, a model reference control approach was proposed for active and passive rehabilitation processes. The system model was tested in a simulation environment, and then all tests were repeated in the physical system. The simulation and real system results include the real system’s open-loop responses, nominal controller responses for each joint, responses, and the results for active, passive, and assistive control modes.