Gene regulatory networks typically have low in-degrees, whereby any given gene is regulated by just a few other genes from a potentially large pool. What mechanisms might be responsible for these low in-degrees? Starting with an accepted framework of the binding of transcription factors to DNA, we consider a simple model of gene regulatory dynamics. In this model, we show that selection leads to the emergence of minimal connectivities compatible with function. We exhibit mathematically this behavior within a simplified limit of our model and show that it also arises in the full model. As a consequence, functionality in these gene networks is parsimonious, i.e., is concentrated on a sparse number of interactions as measured for instance by the lethality of point mutations. Our model thus provides a simple mechanism for the emergence of sparse regulatory networks, and leads to very heterogeneously distributed robustness. Relevant papers: Z. Burda, A. Krzywicki, O. C. Martin, M. Zagorski, Sparse essential interactions in model networks of gene regulation, arXiv: 0910.4077v1 [q-bio.MN]