To meet the dual requirements of precise positioning and passive compliance in variable-stiffness robots, we propose a rigid–elastic–soft coupled DELTA mechanism. The mechanism replaces the traditional DELTA’s active revolute joints with a plug-and-play pneumatic–motor hybrid drive module: the motor actuator provides precise positioning, while the serially connected fabric origami chamber achieves programmable compliance via internal pressure control. Benefiting from the module’s redundant actuation and precisely modelable force–displacement characteristics, the system introduces a 27% increase in variable stiffness without altering its original size or maximum stiffness. Based on a kineto‑static analysis, we established a force‑equilibrium model incorporating external loads, which reveals the differences in output force and stiffness of the fabric origami chamber under varying internal pressures at the same configuration. Furthermore, we developed a stiffness model of the DELTA mechanism under three independently adjustable stiffness actuation units. A prototype was designed and tested, demonstrating that the vertical stiffness at the end effector increases from 4.71 N/mm to 5.99 N/mm as the internal pressure of the chamber increases from 0 kPa to 20 kPa. This work provides a generalizable path for upgrading traditional rigid mechanisms into high-precision, adaptive robots and demonstrates a new paradigm for parallel robots in collaborative manufacturing, precision tasks, and rapid environment switching.