This paper presents a finite element algorithm for solving quasi-static Biot poroelasticity model. By introducing a total pressure, we reformulate the Biot system into a coupled Stokes-parabolic equation. To efficiently solve it, we propose a parallel splitting approach. The coupled system is decomposed into a Stokes subproblem and a parabolic subproblem. These subproblems are then solved in parallel using a stabilization technique. This parallel splitting approach different from sequential or iterative decoupling. The algorithm is proven to be unconditionally stable and theoretical results are validated through numerical experiments.