Three strategies for parallelizing components of the mathematical software package ELLPACK are considered: an explicit approach using compiler directives available only on the target machine, an automatic approach using an optimizing and parallelizing precompiler, and a two-level approach based on extensive use of a set of low level computational kernels. Each approach to parallelization is described in detail, along with a discussion of the effort involved. In connection with the third strategy, a set of computational kernels useful for PDE solving is proposed. We describe our experience in parallelizing six problem solving components of ELLPACK using each of the three strategies and give performance results for a shared memory multiprocessor. Our results suggest that the two-level strategy allows the best balance among programmer effort, portability, and parallel performance.