In this paper we present our efforts to parallelize an unstructured quadrilateral mesh generator. Its serial version is based on the divider-and-conquer idea, and mainly includes two stages, i.e. geometry decomposition and mesh generation. Both stages are parallelized separately. A couple of parallel models are introduced and compared to parallelize the stage of geometry decomposition. A fine-grain level parallel algorithm proves preferable to that based on the task-dependency tree, with which the load imbalance brought by the improper utilization of the symmetry of the vertex pair matrix is removed nicely. Since the number of elements in sub-domains could be pre-computed before meshing, a simple static load balancing scheme is investigated, and the effect of granularity is also discussed briefly. Finally, experiments are designed to evaluate the performance of the parallel mesh generator in detail.