This paper presents the design of a redundant actuated three-DOF Cable-Suspended Parallel Robot (CSPR) with parallelogram architecture and its minimum jerk trajectory planning method. The moving platform of the designed CSPR is supported by eight redundant cables to enable three translational motion degrees. These cables are organized in four sets arranged in parallelogram configurations, with each set driven synchronously by a single motor. To achieve smooth robot motion and avoid abrupt changes in cable tensions, a trajectory planning method is proposed that utilizes seventh degree polynomial curves for minimum jerk in the Cartesian space of the moving platform. A dynamic simulation model of the redundantly actuated CSPR is developed, and point-to-point simulations are conducted. The simulation results indicate that the optimized trajectories exhibit reduced sudden cable tension changes and decreased position tracking errors. Compared to third degree polynomial and seventh degree polynomial trajectory before optimized, the average trajectory errors decreased by 3.6% and 2.3%, respectively, and the maximum trajectory errors are below 3.2 mm.