New parallel computational techniques are introduced for the parallelization of Generic Approximate Sparse Inverse multigrid methods, based on Portable Operating System Interface for UniX (POSIX) threads, for multicore systems. Parallelization of the Generic Approximate Sparse Inverse Matrix (GenAspI) algorithm is achieved based on a new computational approach, namely "strip," which utilizes the data independence of the rows assigned in each available processor. Additionally, new parallel computational techniques are proposed for the parallelization of a modified multigrid V-Cycle method, based on POSIX Threads, for multicore systems. The modified V-Cycle utilized a Parallel PGenAspI Preconditioned Bi-Conjugate Gradient STABilized (BiCGSTAB) as a coarse solver to ensure better parallel performance of the multigrid method. For parallelization purposes, a replication of the multigrid method function is executed on each processor with different index bands and with proper synchronization points to ensure less thread-creation overhead and to maximize parallel performance. Theoretical estimates on speedups and efficiency are also presented. Finally, numerical results for the performance of the PGenAspI algorithm and the PGenAspI---MGV method for solving classical two-dimensional boundary value problems on multicore computer systems are presented. The implementation issues of the proposed method are also discussed using POSIX threads on multicore systems.