The current state-of-the-art dependency parsing approaches employ BiLSTMs to encode input sentences.Motivated by the success of the transformer-based machine translation, this work for the first time applies the self-attention mechanism to dependency parsing as the replacement of the BiLSTM-based encoders, leading to competitive performance on both English and Chinese benchmark data. Based on the detailed error analysis, we then combine the power of both BiLSTM and self-attention via model ensembles, demonstrating their complementary capability of capturing contextual information. Finally, we explore the recently proposed contextualized word representations as extra input features, and further improve the parsing performance.