An elastic rod is a long and thin body able to sustain large global deformations, even if local strains are small. The Cosserat rod is a non-linear elastic rod with an oriented centreline, which enables modelling of bending, stretching and twisting deformations. It can be used for physically-based computer simulation of threads, wires, ropes, as well as flexible surgical instruments such as catheters, guidewires or sutures. We present a massively-parallel implementation of the original CoRdE model as well as our inextensible variation. By superseding the CUDA Scalable Programming Model and using inter-block synchronization, we managed to simulate multiple physics time-steps per single kernel launch utilizing all the GPU's streaming multiprocessors. Under some constraints, this results in nearly constant computation time, regardless of the number of Cosserat elements simulated. When executing 10 time-steps per single kernel launch, our implementation of the original, extensible CoRdE was x40.0 faster. In a number of tests, the GPU implementation of our inextensible CoRdE modification achieved an average speed-up of x15.11 over the corresponding CPU version. Simulating a catheter/guidewire pair (2x512 Cosserat elements) in a cardiovascular application resulted in a 13.5 fold performance boost, enabling for accurate real-time simulation at haptic interactive rates (0.5-1kHz).