With the advent in multicore computers, the scheduling of Grid jobs can be made more effective if scaled to fully utilize the underlying hardware, and parallelized to benefit from the exploitation of multicores. The fact that sequential algorithms do not scale with multicore systems nor benefit from parallelism remains a major obstacle to scheduling in the Grid. As multicore systems become ever more pervasive in our computing lives, over reliance on such systems for passive parallelism does not offer the best option in harnessing the benefits of their multiprocessors for Grid scheduling. An explicit means of exploiting parallelism for Grid scheduling is required. The Group-based Parallel Multi-scheduler, 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 and 8. 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.