Processing massive application graphs on distributed memory systems requires to map the graphs onto the system's processing elements (PEs). This task becomes all the more important when PEs have non-uniform communication costs or the input is highly irregular. Typically, mapping is addressed using partitioning, in a two-step approach or an integrated one. Parallel partitioning tools do exist; yet, corresponding mapping algorithms or their public implementations all have major sequential parts or other severe scaling limitations. In this paper, we propose a parallel algorithm that maps graphs onto the PEs of a hierarchical system. Our solution integrates partitioning and mapping; it models the system hierarchy in a concise way as an implicit labeled tree. The vertices of the application graph are labeled as well, and these vertex labels induce the mapping. The mapping optimization follows the basic idea of parallel label propagation, but we tailor the gain computations of label changes to quickly account for the induced communication costs. Our MPI-based code is the first public implementation of a parallel graph mapping algorithm; to this end, we extend the partitioning library ParHIP. To evaluate our algorithm's implementation, we perform comparative experiments with complex networks in the million- and billion-scale range. In general our mapping tool shows good scalability on up to a few thousand PEs. Compared to other MPI-based competitors, our algorithm achieves the best speed to quality trade-off and our quality results are even better than non-parallel mapping tools.