While neural dependency parsers provide state-of-the-art accuracy for several languages, they still rely on large amounts of costly labeled training data. We demonstrate that in the small data regime, where uncertainty around parameter estimation and model prediction matters the most, Bayesian neural modeling is very effective. In order to overcome the computational and statistical costs of the approximate inference step in this framework, we utilize an efficient sampling procedure via stochastic gradient Langevin dynamics to generate samples from the approximated posterior. Moreover, we show that our Bayesian neural parser can be further improved when integrated into a multi-task parsing and POS tagging framework, designed to minimize task interference via an adversarial procedure. When trained and tested on 6 languages with less than 5k training instances, our parser consistently outperforms the strong BiLSTM baseline (Kiperwasser and Goldberg, 2016). Compared with the BiAFFINE parser (Dozat et al., 2017) our model achieves an improvement of up to 3 for Vietnamese and Irish, while our multi-task model achieves an improvement of up to 9 across five languages: Farsi, Russian, Turkish, Vietnamese, and Irish.