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Parallel continuum robots (PCRs) based on soft actuators have been recently proposed to take advantage of both, soft robots in flexible, diverse actuation, parallel robots in stable, and precise motion. Although various designs have been exhibited, most of them suffer from positioning inaccuracy, especially under uncertain payloads, due to the lack of strong actuation and effective integrated sensing methods. Here, we introduce a vacuum-driven PCR that can simultaneously perform multimode motion, high positioning accuracy, and high load-carrying capacity, on the basis of the mechanical feature of origami. With a soft-rigid hybrid 3-D printing method, the origami linkages of the PCR can be constructed at one time. This forms soft but less stretchable pneumatic chambers that can generate strong actuation based on origami folding. The vacuum-driven linkages exploit the contraction-twisting coupled folding characteristic of the Kresling origami pattern. The length of each linkage can be determined by recording the twisting angles. Theoretical models for both the self-sensing linkage and the PCR with three individually actuated linkages, as well as a closed-loop feedback control strategy, have been presented for the motion control of the PCR. The experimental results of a PCR prototype demonstrate its multimode motion, including contraction/extension, omnidirectional bending, and circular swing. The combination of the design and fabrication methods, the sensing strategy, and the feedback control enables the prototype to perform high positioning accuracy with various trajectories, even under a 2-kg payload. The design concept and comprehensive guidelines in this work aim to expand the capacities of PCRs or soft robots for broader applications, and facilitate the usage of origami and multimaterial 3-D printing in building robotic structures.