We introduce UniRST, the first unified RST-style discourse parser capable of handling 18 treebanks in 11 languages without modifying their relation inventories. To overcome inventory incompatibilities, we propose and evaluate two training strategies: Multi-Head, which assigns separate relation classification layer per inventory, and Masked-Union, which enables shared parameter training through selective label masking. We first benchmark monotreebank parsing with a simple yet effective augmentation technique for low-resource settings. We then train a unified model and show that (1) the parameter efficient Masked-Union approach is also the strongest, and (2) UniRST outperforms 16 of 18 mono-treebank baselines, demonstrating the advantages of a single-model, multilingual end-to-end discourse parsing across diverse resources.