Explicit finite element approximate inverse preconditioning methods have been extensively used for solving efficiently sparse linear systems on multiprocessor and multicomputer systems. New parallel computational techniques are proposed for the parallelization of explicit preconditioned biconjugate conjugate gradient type methods, based on Portable Operating System Interface for UniX (POSIX) Threads, for multicore systems. Parallelization is achieved by assigning every loop of the parallel explicit preconditioned bi-conjugate conjugate gradient-STAB (PEPBiCG-STAB) to the desired number of threads, thus achieving for-loop parallelization. Theoretical estimates on speedups and efficiency are also presented. Finally, numerical results for the performance of the PEPBiCG-STAB method for solving characteristic 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 a multicore system.