Genomic data is growing rapidly due to the high demand for precision medicine. Consequently, efficient genome compression algorithms are needed to reduce storage usage in an acceptable response time. This paper introduces HybridHRCM, a hybrid MPI/OpenMP algorithm that harnesses the power of multicore clusters to compress a collection of genomic sequences. We compared our proposal with MtHRCM-opt, its multi-threaded OpenMP counterpart that is suitable for single-node multicore systems. Experimental results demonstrate that HybridHRCM enhances the scalability of MtHRCM-opt for large test collections when using the same number of cores but in a distributed way, while behaves similarly for small test collections. Furthermore, the results reveal that HybridHRCM still achieving good performance when adding more nodes, for all collections.