A strong inductive bias is essential in unsupervised grammar induction. In this paper, we explore a particular sparsity bias in dependency grammars that encourages a small number of unique dependency types. We use part-of-speech (POS) tags to group dependencies by parent-child types and investigate sparsity-inducing penalties on the posterior distributions of parent-child POS tag pairs in the posterior regularization (PR) framework of Graça et al. (2007). In experiments with 12 different languages, we achieve significant gains in directed attachment accuracy over the standard expectation maximization (EM) baseline, with an average accuracy improvement of 6.5%, outperforming EM by at least 1% for 9 out of 12 languages. Furthermore, the new method outperforms models based on standard Bayesian sparsity-inducing parameter priors with an average improvement of 5% and positive gains of at least 1% for 9 out of 12 languages. On English text in particular, we show that our approach improves performance over other state-of-the-art techniques.