We propose a parallel algorithm for computing the power series solutions to an arbitrary linear ordinary differential equation with power series coefficients at one of its ordinary points. We take advantage of a tiling strategy. Our experimental results show that the proposed algorithm achieves significant speedup factors with respect to its serial counterpart as well as to other serial implementation solving the same problem. The code we developed is part of the Basic Polynomial Algebra Subprograms (BPAS) and is publicly available.