This paper presents a climbing robot (CR) designed for the purpose of pipeline maintenance, with capability to avoid the risks inherent in manual operations. In the design process, a three degree of freedom (DOF) parallel mechanism coupled with a remote center of motion (RCM) mechanism linkage mechanism were designed to serve as the CR’s climbing mechanism, which met the specific demands for climbing movements. The modified Kutzbach–Grübler formula and the screw theory were applied to calculate the DOFs of the CR. Then, the inverse and forward position analysis for the CR was derived. Furthermore, velocity and acceleration analysis of parallel mechanism were conducted and derived the Jacobian matrix, through which the singularity of parallel mechanism was analyzed. In order to evaluate kinematic performance of parallel mechanism, the motion/force transmission index (LTI) of workspace was calculated, which directed the followed dimensional optimization process. According to the optimization result, a prototype was constructed and a series of motion experiments were carried out to validate its climbing capability.