We propose a parallel massage robot with compliant joints based on the series elastic actuator (SEA), offering a unified force-position control approach. First, the kinematic and static force models are established for obtaining the corresponding control variables. Then, a novel force-position control strategy is proposed to separately control the force-position along the normal direction of the surface and another two-direction displacement, without the requirement of a robotic dynamics model. To evaluate its performance, we implement a series of robotic massage experiments. The results demonstrate that the proposed massage manipulator can successfully achieve desired forces and motion patterns of massage tasks, arriving at a high-score user experience.