The semantic segmentation of parts of objects in the wild is a challenging\ntask in which multiple instances of objects and multiple parts within those\nobjects must be detected in the scene. This problem remains nowadays very\nmarginally explored, despite its fundamental importance towards detailed object\nunderstanding. In this work, we propose a novel framework combining higher\nobject-level context conditioning and part-level spatial relationships to\naddress the task. To tackle object-level ambiguity, a class-conditioning module\nis introduced to retain class-level semantics when learning parts-level\nsemantics. In this way, mid-level features carry also this information prior to\nthe decoding stage. To tackle part-level ambiguity and localization we propose\na novel adjacency graph-based module that aims at matching the relative spatial\nrelationships between ground truth and predicted parts. The experimental\nevaluation on the Pascal-Part dataset shows that we achieve state-of-the-art\nresults on this task.\n