We present a deep neural architecture that parses sentences into three semantic dependency graph formalisms. By using efficient, nearly arc-factored inference and a bidirectional-LSTM composed with a multi-layer perceptron, our base system is able to significantly improve the state of the art for semantic dependency parsing, without using hand-engineered features or syntax. We then explore two multitask learning approaches—one that shares parameters across formalisms, and one that uses higher-order structures to predict the graphs jointly. We find that both approaches improve performance across formalisms on average, achieving a new state of the art. Our code is open-source and available at https://github.com/Noahs-ARK/NeurboParser.