Phylogenetic networks provide a powerful framework for modeling and analyzing reticulate evolutionary histories. While polyploidy has been shown to be prevalent not only in plants but also in other groups of eukaryotic species, most work done thus far on phylogenetic network inference assumes diploid hybridization. These inference methods have been applied, with varying degrees of success, to data sets with polyploid species, even though polyploidy violates the mathematical assumptions underlying these methods. Statistical methods were developed recently for handling specific types of polyploids and so were parsimony methods that could handle polyploidy more generally yet while excluding processes such as incomplete lineage sorting. In this article, we introduce a new method for inferring most parsimonious phylogenetic networks on data that include polyploid species. Taking gene tree topologies as input, the method seeks a phylogenetic network that minimizes deep coalescences while accounting for polyploidy. We demonstrate the performance of the method on both simulated and biological data. The inference method as well as a method for evaluating evolutionary hypotheses in the form of phylogenetic networks are implemented and publicly available in the PhyloNet software package. [Incomplete lineage sorting; minimizing deep coalescences; multilabeled trees; multispecies network coalescent; phylogenetic networks; polyploidy.]