In this paper, we extend a generic library framework based on the state monad to exploit deterministic parallelism in a purely functional language Haskell and provide benchmarks for the extended features on a multicore machine. Although purely functional programs are known to be well-suited to exploit parallelism, unintended squential data dependencies could prohibit effective parallelism. Symbolic execution programs usually implement fresh name generation in order to prevent confusion between variables in different scope with the same name. Such implementations are often based on squential state management, working against parallelism. We provide reusable primitives to help developing parallel symbolic execution programs with unbound-genercis, a generic name-binding library for Haskell, avoiding sequential dependencies in fresh name generation. Our parallel extension does not modify the internal implementation of the unbound-generics library, having zero possibility of degrading existing serial implementations of symbolic execution based on unbound-genecrics. Therefore, our extension can be applied only to the parts of source code that need parallel speedup.