This paper focuses on configuration design, dimensional synthesis, and engineering application of a novel asymmetric 2R1T parallel mechanism (PM) with zero-coupling degree. The analytical forward and inverse displacement solutions are deduced by the means of vector method. The mathematical models between Euler angles and the orientational parameters (i.e., azimuth and tilt angles) of the offset output axis are established. Using screw theory as mathematical tool, this paper worked out evaluation indices of motion/force transmissibility and presented the definitions and calculation methods of good transmission orientation workspace and good transmission orientation capacity (GTOC). Furthermore, a comparative example with respect to kinematic performance of asymmetric UPS-RPU-PU PM and planar symmetric 2UPS-PU PM is carried out, and the result demonstrates that UPS-RPU-PU significantly outperforms 2UPS-PU in terms of GTOC. The constrained optimization model is constructed to formulate optimal problem of dimensional parameters on the maximizing GTOC, which is then solved by differential evolution algorithm. Finally, an engineering case demonstrates that the optimized mechanism has a good application prospect in hydraulic support test bed.