Discourse Representation Structure (DRS) is an innovative semantic representation designed to capture the meaning of texts with arbitrary lengths across languages. The semantic representation parsing is essential for achieving natural language understanding through logical forms. Nevertheless, the performance of DRS parsing models remains constrained when trained exclusively on monolingual data. To tackle this issue, we introduce a cross-lingual training strategy. The proposed method is model-agnostic yet highly effective. It leverages cross-lingual training data and fully exploits the alignments between languages encoded in pre-trained language models. The experiments conducted on the standard benchmarks demonstrate that models trained using the cross-lingual training method exhibit significant improvements in DRS clause and graph parsing in English, German, Italian and Dutch. Comparing our final models to previous works, we achieve state-of-the-art results in the standard benchmarks. Furthermore, the detailed analysis provides deep insights into the performance of the parsers, offering inspiration for future research in DRS parsing. We keep updating new results on benchmarks to the appendix.