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Open-vocabulary part segmentation (OVPS) struggles with structurally connected boundaries due to the inherent conflict between continuous image features and discrete classification mechanism. To address this, we propose PBAPS, a novel training-free framework specifically designed for OVPS. PBAPS leverages structural knowledge of object-part relationships to guide a progressive segmentation from objects to fine-grained parts. To further improve accuracy at challenging boundaries, we introduce a Boundary-Aware Refinement (BAR) module that identifies ambiguous boundary regions by quantifying classification uncertainty, enhances the discriminative features of these ambiguous regions using high-confidence context, and adaptively refines part prototypes to better align with the specific image. Experiments on Pascal-Part-116, ADE20K-Part-234, PartImageNet demonstrate that PBAPS significantly outperforms state-of-the-art methods, achieving 46.35\% mIoU and 34.46\% bIoU on Pascal-Part-116. Our code is available at https://github.com/TJU-IDVLab/PBAPS.