Time-optimal trajectory planning (TOTP) is a well-studied problem in robotics and manufacturing, which involves the minimization of the time required for the operation point of a mechanism to follow a path, subject to constraints on velocity, acceleration, jerk, torque, and other parameters. A TOTP technique, designed for fully-specified paths that include abrupt changes in direction, was previously introduced by the authors: an incremental approach called minimum-time trajectory shaping (MTTS) was used, which gradually shapes the temporal and spatial parameters of the trajectory to produce a near-optimal solution. In the current paper, MTTS is adapted for use with cable-driven parallel mechanisms, which exhibit the additional constraint that all cable tensions remain positive along a path to be followed. Typically, minimum cable tension is imposed indirectly for a cable-driven parallel mechanism: an iterative technique is used whereby parameters, such as the maximum acceleration, are adjusted until the minimum tension along the trajectory, computed a posteriori, is barely above a minimum threshold. This technique is time-consuming for the trajectory programmer, and is not time-optimal because the entire trajectory is affected by the global adjustment of parameters. For the new technique proposed in this paper, the minimum-tension constraint is imposed directly and is fully integrated with MTTS during trajectory generation, thus maintaining a time-optimal solution. This approach is relevant for cable-driven mechanism applications that involve high accelerations, particularly in the vertical direction. It could also be useful for applications that include paths near the static workspace boundary for a mechanism, where tension constraints typically dominate.