In this paper we discuss the parallel implementation of the Cholesky factorization of a positive definite symmetric matrix when that matrix is block tridiagonal. While parallel implementations for this problem, and closely related problems like the factorization of banded matrices, have been previously reported in the literature, those implementations dealt with the special cases where the block size (bandwidth) was either very large (wide) or very small (narrow). We present a solution that can be used for the entire spectrum of cases, ranging from extremely large (wide) to very small (narrow). Preliminary performance results collected on a Cray T3E-600 distributed memory supercomputer show that our implementation attains respectable performance. Indeed, factorization of a matrix with block size b = 1000 and a total dimension of more than 500,000 takes about 3.6 minutes on 128 processors.