AbstractGraph-based semantic parsing aims to represent textual meaning through directed graphs. As one of the most promising general-purpose meaning representations, these structures and their parsing have gained a significant interest momentum during recent years, with several diverse formalisms being proposed. Yet, owing to this very heterogeneity, most of the research effort has focused mainly on solutions specific to a given formalism. In this work, instead, we reframe semantic parsing towards multiple formalisms as Multilingual Neural Machine Translation (MNMT), and propose SGL, a many-to-many seq2seq architecture trained with an MNMT objective. Backed by several experiments, we show that this framework is indeed effective once the learning procedure is enhanced with large parallel corpora coming from Machine Translation: we report competitive performances on AMR and UCCA parsing, especially once paired with pre-trained architectures. Furthermore, we find that models trained under this configuration scale remarkably well to tasks such as cross-lingual AMR parsing: SGL outperforms all its competitors by a large margin without even explicitly seeing non-English to AMR examples at training time and, once these examples are included as well, sets an unprecedented state of the art in this task. We release our code and our models for research purposes at https://github.com/SapienzaNLP/sgl.