According to the topology design theory and method of parallel mechanism (PM) based on position and orientation characteristic (POC) equations, a PM which can realize one-translation and two-rotation (1T2R) was designed. The main topological characteristics of the PM, such as POC, degree of freedom (DOF) and coupling degree (κ) were analyzed. It was shown that the PM contained only one sub-kinematic chain (SKC) and its coupling degree was one, but its symbolic direct position solutions could not be obtained. Only the numerical methods could be used to solve its numerical direct position solutions. Therefore, the topology coupling-reducing design was carried out to obtain the 1T2R PM with zero coupling degree, symbolic direct position solutions but constant POC and DOF, because zero coupling degree would benefit the subsequent error analysis, trajectory planning, motion control and dynamic analysis of the PM. Both the symbolic direct position solutions and inverse solutions were derived, and the correctness of them were proved by example. The singularities of the PM were analyzed based on the Jacobian matrix, and some cases of the singular positions of the PM were given by the graphs. Based on symbolic direct position solutions, the workspace of the PM was calculated and analyzed, which was consistent with that obtained by the inverse position solutions. At the same time, it showed that the workspace of the PM was large and the shape was regular. It was found that the workspace based on symbolic direct position solutions had the advantages of less computation and accurate boundaries.