Disability from injuries and diseases is a global problem affecting a large population; however, due to a lack of therapists and labor-intensive procedures, only a few benefits from rehabilitation. Robots can assist therapists in treating many patients simultaneously, but the existing solutions need improvements in their mechanism, actuation, and control. This article presents a four-link parallel end-effector robot for wrist joint rehabilitation. The proposed robot employs biomimetic muscle actuators (BMA) that provide intrinsic compliance to the robotic system. A fuzzy-based model is developed to identify the nonlinear nature of BMAs. The stiffness-observer learns subject-specific stiffness, which is used to modify the robot reference trajectories. An adaptive controller uses the fuzzy model and stiffness-observer and simultaneously controls the four BMAs to provide three degrees of rotational freedom to the robot end-effector. The feasibility of the robot mechanism and the controller was evaluated through proof of concept experiments conducted with three unimpaired human subjects. It was found that the controller was able to guide the robot-human system on the commanded trajectories in the presence of parallel actuation of compliant and nonlinear BMAs. Furthermore, the controller was also able to modify the commanded trajectories in the higher stiffness regions of the wrist workspace.