The dataflow model allows natural expression of parallelism in an application. Applications expressed in the dataflow model can be executed either using the data-driven or the demand-driven schemes. Although both these schemes have their utility in different scenarios, the realization of the demand-driven scheme is not adequately supported in the existing solutions for task parallelism. In this paper, we examine some of the requirements placed by the demand-driven execution scheme on task parallelism. We present PFunc, a new library-based solution for task parallelism that fully supports the demand-driven execution scheme. We compare the runtimes and peak memory consumption of an unsymmetric sparse LU factorization emulation parallelized using both the data- and demand-driven execution schemes. This comparison shows that the demand-driven model provides benefits that necessitate its full support in task parallelism.