The Group-based Parallel Multi-scheduler (GPMS), introduced in this paper, is aimed at effectively exploiting the benefits of multicore systems for Grid scheduling by splitting jobs and machines into paired groups and independently scheduling jobs in parallel from those groups. We implemented two job grouping methods; Execution Time Balanced (ETB) and Execution Time Sorted then Balanced (ETSB), and two machine grouping methods,; Evenly Distributed (EvenDist) and Similar Together (SimTog). For each method, we varied the number of groups between 2, 4, 8 and 16. We then executed the MinMin Grid scheduling algorithm independently within the groups. We demonstrated that by sharing jobs and machines into groups before scheduling, the computation time for the scheduling process drastically improved by magnitudes of 85% over the ordinary MinMin algorithm when implemented on a HPC system. We also found that our balanced group based approach achieved better results than our previous Priority based grouping approach.