Recent advances in the field of multilingual dependency parsing have brought the idea of a truly universal parser closer to reality. However, cross-language interference and restrained model capacity remain a major obstacle to this pursuit. To address these issues, we propose a novel multilingual task adaptation approach based on recent work in parameter-efficient transfer learning, which allows for an easy but effective integration of existing linguistic typology features into the parsing network. The resulting parser, UDapter, consistently outperforms strong monolingual and multilingual baselines on both high-resource and low-resource (zero-shot) languages, setting a new state of the art in multilingual UD parsing. Our in-depth analyses show that soft parameter sharing via typological features is key to this success.