Phylogenies play a major role in representing the interrelationships among biological entities. Many methods for reconstructing and studying such phylogenies have been proposed, almost all of which assume that the underlying history of a given set of species can be represented by a binary tree. Although many biological processes can be effectively modeled and summarized in this fashion, others cannot: recombination, hybrid speciation, and horizontal gene transfer result in networks, rather than trees, of relationships. In a series of papers, we have extended the maximum parsimony (MP) criterion to phylogenetic networks, demonstrated its appropriateness, and established the intractability of the problem of scoring the parsimony of a phylogenetic network. In this work we show the hardness of approximation for the general case of the problem, devise a very fast (linear-time) heuristic algorithm for it, and implement it on simulated as well as biological data.