Closed streamlines are an integral part of vector field topology, since they behave like sources respectively sinks but are often neither considered nor detected. If a streamline computation makes too many steps or takes too long, the computation is usually terminated without any answer on the final behavior of the streamline. We developed an algorithm that detects closed streamlines during the integration process. Since the detection of all closed streamlines in a vector field requires the computation of any streamlines we extend this algorithm to a parallel version to enhance computational speed. To test our implementation we use a numerical simulation of a swirling jet with an inflow into a steady medium. We built two different Linux clusters as parallel test systems where we check the performance increase when adding more processors to the cluster. We show that we have a very low parallel overhead due to the neglectable communication expense of our implementation.