Real-world objects are composed of distinctive, object-specific parts. Identifying these parts is key to performing fine-grained, compositional reasoning-yet, large multimodal models (LMMs) struggle to perform this seemingly straightforward task. In this work, we introduce PARTONOMY, an LMM benchmark designed for pixel-level part grounding. We construct PARTONOMY from existing part datasets and our own rigorously annotated set of images, encompassing 862 part labels and 534 object labels for evaluation. Unlike existing datasets that simply ask models to identify generic parts, PARTONOMY uses specialized concepts (e.g., agricultural airplane), and challenges models to compare objects' parts, consider part-whole relationships, and justify textual predictions with visual segmentations. Our experiments demonstrate significant limitations in state-of-the-art LMMs (e.g., LISA-13B achieves only 5.9% gIoU), highlighting a critical gap in their part grounding abilities. We note that existing segmentation-enabled LMMs (segmenting LMMs) have two key architectural shortcomings: they use special [SEG] tokens not seen during pretraining which induce distribution shift, and they discard predicted segmentations instead of using past predictions to guide future ones. To address these deficiencies, we train several part-centric LMMs and propose PLUM, a novel segmenting LMM that uses span tagging instead of segmentation tokens and that conditions on prior predictions in a feedback loop. We find that pretrained PLUM outperforms existing segmenting LMMs on reasoning segmentation, VQA, and visual hallucination benchmarks. In addition, PLUM finetuned on our proposed Explanatory Part Segmentation task is competitive with segmenting LMMs trained on significantly more segmentation data. Our work opens up new avenues towards enabling fine-grained, grounded visual understanding in LMMs.