We introduce a novel dependency parser, the hexatagger, that constructs dependency trees by tagging the words in a sentence with elements from a finite set of possible tags. In contrast to many approaches to dependency parsing, our approach is fully parallelizable at training time, i.e., the structure-building actions needed to build a dependency parse can be predicted in parallel to each other. Additionally, exact decoding is linear in time and space complexity. Furthermore, we derive a probabilistic dependency parser that predicts hexatags using no more than a linear model with features from a pretrained language model, i.e., we forsake a bespoke architecture explicitly designed for the task. Despite the generality and simplicity of our approach, we achieve state-of-the-art performance of 96.4 LAS and 97.4 UAS on the Penn Treebank test set. Additionally, our parser’s linear time complexity and parallelism significantly improve computational efficiency, with a roughly 10-times speed-up over previous state-of-the-art models during decoding.