The work aims to analyze whether a three-degree-of-freedom 2-UPS/(S+SPR)R parallel mechanism with closed-loop elements has concomitant motion and its forward and inverse position solutions. The rotation matrix was obtained by Euler's change, and the constraint equation was established based on the structural characteristics of the mechanism to analyze whether the proposed mechanism had concomitant motion and its reverse position solution. The particle swarm optimization (PSO) algorithm was used to analyze the forward position of the mechanism. The mechanism had no concomitant motion of the z-axis rotation, and the inverse position equation was established. The PSO algorithm could accurately obtain the position of the moving platform when the driving parameters were entered. The proposed mechanism does not have concomitant motion. The precise numerical solution of the forward position solution is obtained through the PSO optimization algorithm, which provides a good basis for analyzing the working space of the mechanism.