The carbon footprint of natural language processing research has been increasing in recent years due to its reliance on large and inefficient neural network implementations. Distillation is a network compression technique which attempts to impart knowledge from a large model to a smaller one. We use teacher-student distillation to improve the efficiency of the Biaffine dependency parser which obtains state-of-the-art performance with respect to accuracy and parsing speed (Dozat and Manning, 2017). When distilling to 20\% of the original model's trainable parameters, we only observe an average decrease of $\sim$1 point for both UAS and LAS across a number of diverse Universal Dependency treebanks while being 2.30x (1.19x) faster than the baseline model on CPU (GPU) at inference time. We also observe a small increase in performance when compressing to 80\% for some treebanks. Finally, through distillation we attain a parser which is not only faster but also more accurate than the fastest modern parser on the Penn Treebank.