We describe a parallel algorithm for finding the fill that occurs when a sparse symmetric positive definite matrix A is factored into its Cholesky factor L. The algorithm is in two steps: First we determine the elimination forest F for A. Then from F and A we compute the fill. The algorithm takes $O(\log^{2} n)$ time, using $m + n$ processors to find the elimination forest and $m^{*}+ n$ processors to find the fill.