We present substructure distribution projection (SubDP), a technique that projects a distribution over structures in one domain to another, by projecting substructure distributions separately. Models for the target domain can then be trained, using the projected distributions as soft silver labels. We evaluate SubDP on zero shot cross-lingual dependency parsing, taking dependency arcs as substructures: we project the predicted dependency arc distributions in the source language(s) to target language(s), and train a target language parser on the resulting distributions. Given an English tree bank as the only source of human supervision, SubDP achieves better unlabeled attachment score than all prior work on the Universal Dependencies v2.2 (Nivre et al., 2020) test set across eight diverse target languages, as well as the best labeled attachment score on six languages. In addition, SubDP improves zero shot cross-lingual dependency parsing with very few (e.g., 50) supervised bitext pairs, across a broader range of target languages.