Parallel mechanism is characterized by balanced load bearing and compact layout, which is widely used in aviation, aerospace, manufacturing, robots and other fields. Proper driving and control strategies are very important to improve the bearing and motion performance of the parallel mechanism. The controllable drive selection is the primary research for controlling the parallel mechanism. In this paper, a new parallel mechanism with various motion modes is proposed. To analyze the motion and constraint contributions from the motion pairs to the whole mechanism, all the motion pairs in the mechanism are assumed to be flexible joints according to the linear stiffness superposition model. Then the mathematical condition for the controllable drive selection is obtained. In addition, the research finds that the mechanism has two special motion modes, pure rotation and pure translation, which has reference significance for designing the motion mode of the parallel mechanism.