The multilevel heuristic is an effective strategy for speeding up graph analytics, and graph coarsening is an integral step of multilevel methods. We perform a comprehensive study of multilevel coarsening in this work. We primarily focus on the graphics processing unit (GPU) parallelization of the Heavy Edge Coarsening (HEC) method executed in an iterative setting. We present optimizations for the two phases of coarsening, a fine-to-coarse vertex mapping phase, and a coarse graph construction phase. We also express several other coarsening algorithms using the Kokkos framework and discuss their parallelization. We demonstrate the efficacy of parallelized HEC on an NVIDIA Turing GPU and a 32-core AMD Ryzen processor using multilevel spectral graph partitioning as the primary case study.