We present algorithms for higher-order dependency parsing that are "third-order" in the sense that they can evaluate substructures containing three dependencies, and "efficient" in the sense that they require only O(n4) time. Importantly, our new parsers can utilize both sibling-style and grandchild-style interactions. We evaluate our parsers on the Penn Treebank and Prague Dependency Treebank, achieving unlabeled attachment scores of 93.04% and 87.38%, respectively.