A software framework taking advantage of parallel processing capabilities of CPUs and GPUs is designed for the real‐time interactive cutting simulation of deformable objects. Deformable objects are modelled as voxels connected by links. The voxels are embedded in an octree mesh used for deformation. Cutting is performed by disconnecting links swept by the cutting tool and then adaptively refining octree elements near the cutting tool trajectory. A surface mesh used for visual display is reconstructed from disconnected links using the dual contour method. Spatial hashing of the octree mesh and topology‐aware interpolation of distance field are used for collision. Our framework uses a novel GPU implementation for inter‐object collision and object self collision, while tool‐object collision, cutting and deformation are assigned to CPU, using multiple threads whenever possible. A novel method that splits cutting operations into four independent tasks running in parallel is designed. Our framework also performs data transfers between CPU and GPU simultaneously with other tasks to reduce their impact on performances. Simulation tests show that when compared to three‐threaded CPU implementations, our GPU accelerated collision is 53–160% faster; and the overall simulation frame rate is 47–98% faster.