Coarse-to-fine inference has been shown to be a robust approximate method for improving the efficiency of structured prediction models while preserving their accuracy. We propose a multi-pass coarse-to-fine architecture for dependency parsing using linear-time vine pruning and structured prediction cascades. Our first-, second-, and third-order models achieve accuracies comparable to those of their unpruned counterparts, while exploring only a fraction of the search space. We observe speed-ups of up to two orders of magnitude compared to exhaustive search. Our pruned third-order model is twice as fast as an unpruned first-order model and also compares favorably to a state-of-the-art transition-based parser for multiple languages.