Modern large-scale scientific applications consist of thousands to millions of individual tasks. These tasks involve not only computation but also communication with one another. Typically, the communication pattern between tasks is sparse and can be determined in advance. Such applications are executed on supercomputers, which are often organized in a hierarchical hardware topology, consisting of islands, racks, nodes, and processors, where processing elements reside. To ensure efficient workload distribution, tasks must be allocated to processing elements in a way that ensures balanced utilization. However, this approach optimizes only the workload, not the communication cost of the application. It is straightforward to see that placing groups of tasks that frequently exchange large amounts of data on processing elements located near each other is beneficial. The problem of mapping tasks to processing elements considering optimization goals is called process mapping. In this work, we focus on minimizing communication cost while evenly distributing work. We present the first shared-memory algorithm that utilizes hierarchical multisection to partition the communication model across processing elements. Our parallel approach achieves the best solution on 95 percent of instances while also being marginally faster than the next best algorithm. Even in a serial setting, it delivers the best solution quality while also outperforming previous serial algorithms in speed.