The current leading algorithms for convolutional sparse coding are not inherently parallelizable, and therefore are not able to fully exploit modern multi-core architectures. We address this deficiency by developing a new algorithm that partitions the dictionary and the corresponding coefficient maps into groups, solving the main subproblems for all of the groups in parallel. Theoretical complexities and implementational details are discussed and validated with computational experiments, which indicate speed improvements by about a factor of 5, depending on the specific problem.