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Open-vocabulary part segmentation (OVPS) is an emerging research area focused on segmenting fine-grained entities based on diverse and previously unseen vocabularies. Our study highlights the inherent complexities of part segmentation due to intricate boundaries and diverse granularity, reflecting the knowledge-based nature of part identification. To address these challenges, we propose PartCLIPSeg, a novel framework utilizing generalized parts and object-level contexts to mitigate the lack of generalization in fine-grained parts. PartCLIPSeg integrates competitive part relationships and attention control techniques, alleviating ambiguous boundaries and underrepresented parts. Experimental results demonstrate that PartCLIPSeg outperforms existing state-of-the-art OVPS methods, offering refined segmentation and an advanced understanding of part relationships in images. Through extensive experiments, our model demonstrated an improvement over the state-of-the-art models on the Pascal-Part-116, ADE20K-Part-234, and PartImageNet datasets.