Higher order derivative tensors are efficiently computed via univariate Taylor polynomials in multiple directions. This permits parallelization across the directions. We compare various parallelization approaches in the context of generated overloading libraries. In particular we compare the use of OpenMP vs another approach that uses pthreads with atomic operations provided by MCS's own OPA library. We discuss the bottlenecks and the overhead in the current implementation in the Rapsodia tool.