A note on parallel finite difference approximate inverse preconditioning on multicore systems using POSIX threads. New parallel computational techniques are introduced for the parallelization of explicit finite difference (FD) approximate inverse matrix methods, based on Portable Operating System Interface for UniX (POSIX) threads, for multicore systems. Parallelization of the Optimized Banded Generalized Approximate Inverse Matrix (OBGAIM) algorithm is achieved based on the concept of the ”fish bone” approach with the use of a thread pool pattern. Theoretical estimates on speedups and efficiency are also presented. Additionally, new parallel computational techniques are proposed for the parallelization of explicit preconditioned biconjugate conjugate gradient type methods, based on POSIX threads, for multicore systems. For parallelization purposes a replication of the parallel explicit preconditioned biconjugate conjugate gradient-STAB (PEPBICG-STAB) method was assigned on each created thread, with different index bands and with proper synchronization points on inner products and matrix-vector multiplications. Theoretical estimates on speedups and efficiency are also presented. Finally, numerical results for the performance of the Parallel Fish Bone OBGAIM (PaFiBo-OBGAIM) algorithm and the PEPBICG-STAB method for solving classical two-dimensional boundary value problems on multicore computer systems are presented, which are favorably compared to corresponding results from multiprocessor systems. The implementation issues of the proposed method are also discussed using POSIX threads on multicore systems.