Graphs capture the essential elements of many problems broadly defined as searching or categorizing. With the rapid increase of data volumes from sensors, many application disciplines need to process larger graphs quickly. This paper presents the results of parallelizing with OpenMP an algorithm that finds, in a single large labeled undirected sparse graph, the connected subgraphs with a given minimum number of edge-disjoint embeddings. Parallelism is exploited at two levels in the algorithm. The lack of a priori knowledge of the extent of parallelism for a given input required use of a dynamic, multi-level approach based on the proposed OpenMP taskq/task extensions. The parallel implementation required the addition of 21 directives and about 50 accompanying lines of code, in an original code of about 15,000 lines. Experimental results show excellent speed-up to 30 processors for the graphs used, with a best speed-up of 26.1 compared to the serial version. The taskq/task constructs show promise for problems exhibiting unstructured parallelism.