Many two-dimensional incompressible inviscid vortex flows can be simulated very efficiently by means of the contour dynamics method. Several applications require the use of a hierarchical-element method (HEM), which is a modified version of the classical contour dynamics scheme based on the fast multipole method. The HEM can be used, for example, to study the large-scale motion of coherent structures in idealized geophysical fluid dynamics where the flow can be modelled as the motion in a thin layer of fluid in the presence of a non-uniform background rotation. Nevertheless, such simulations require a substantial computational effort, even when the HEM is used. In this article it is shown that the acceleration of contour dynamics simulations can be increased further by parallelizing the HEM algorithm. Speed-up, load balance and scalability are parallel performance features which are studied for several representative cases. The HEM has been parallelized using the OpenMP interface and is tested with up to 16 processors on an Origin 3800 CC-NUMA computer.