PU- and P*U*-equivalent parallel mechanisms (PMs) are critical families of PMs with one translational and two rotational (1T2R) degree of freedoms and have always been a research hotspot among lower mobility PMs. However, researches on these two families of PMs remain to be inadequate, and existing types are few and uncomprehensive. In this study, first, general wrench systems of PU-equivalent are derived based on virtual-chain approach and screw theory, revealing its constraint characteristics under general configuration. Cause of one parasitic motion is put forward and general wrench systems of P*U*-equivalent PMs with one parasitic motion are obtained. In addition, constraint analysis has been carried out to figure out constraint characteristics of P*U*-equivalent PMs with three parasitic motions. Second, branch chains are divided by generating constraints, and their structures are synthesized based on the presented rules. Then, the process for type synthesis of PU-equivalent PMs and P*U*-equivalent PMs and a series of novel 1T2R PMs are attained based on this. Finally, a novel PU-equivalent PM, 2PRU-PRUPc, and a novel P*U*-equivalent PM with one parasitic motion, 2PRU-PUU PM, are analyzed, demonstrating the effectiveness of the proposed type synthesis method.