This paper is concerned with the solution of block tridiagonal linear algebraic systems by the preconditioned conjugate gradient (PCG) method. If we consider two splittings of the coefficient matrix, it is possible to derive a parallel additive polynomial preconditioner and to give conditions for such preconditioner to be symmetric positive definite. For the diffusion problem this preconditioner can be interpreted as a simple form of domain decomposition preconditioning. In order to solve each subdomain problem we analyse two solvers named Cyclic Reduction solver and Approximate Schur solver. Numerical results carried out on Cray Y-MP for a set of test-problems permit to evaluate the effectiveness of the parallel polynomial preconditioner.