We study the problem of analyzing tweets with universal dependencies (UD). We extend the UD guidelines to cover special constructions in tweets that affect tokenization, part-of-speech tagging, and labeled dependencies. Using the extended guidelines, we create a new tweet treebank for English (Tweebank v2) that is four times larger than the (unlabeled) Tweebank v1 introduced by Kong et al. (2014). We characterize the disagreements between our annotators and show that it is challenging to deliver consistent annotation due to ambiguity in understanding and explaining tweets. Nonetheless, using the new treebank, we build a pipeline system to parse raw tweets into UD. To overcome the annotation noise without sacrificing computational efficiency, we propose a new method to distill an ensemble of 20 transition-based parsers into a single one. Our parser achieves an improvement of 2.2 in LAS over the un-ensembled baseline and outperforms parsers that are state-of-the-art on other treebanks in both accuracy and speed.