We present ParHDE, a shared-memory parallelization of the High-Dimensional Embedding (HDE) graph algorithm. Originally proposed as a graph drawing algorithm, HDE characterizes the global structure of a graph and is closely related to spectral graph computations such as computing the eigenvectors of the graph Laplacian. We identify compute- and memory-intensive steps in HDE and parallelize these steps for efficient execution on shared-memory multicore platforms. ParHDE can process graphs with billions of edges in minutes, is up to 18 × faster than a prior parallel implementation of HDE, and achieves up to a 24 × relative speedup on a 28-core system. We also implement several extensions of ParHDE and demonstrate its utility in diverse graph computation-related applications.