Part-level 3D generation is essential for applications requiring decomposable and structured 3D synthesis. However, existing methods either rely on implicit part segmentation with limited granularity control or depend on strong external segmenters trained on large annotated datasets. In this work, we observe that part awareness emerges naturally during whole-object geometry learning and propose Geom-Seg VecSet, a unified geometry-segmentation latent representation that jointly encodes object geometry and part-level structure. Building on this representation, we introduce UniPart, a two-stage latent diffusion framework for image-guided part-level 3D generation. The first stage performs joint geometry generation and latent part segmentation, while the second stage conditions part-level diffusion on both whole-object and part-specific latents. A dual-space generation scheme further enhances geometric fidelity by predicting part latents in both global and canonical spaces. Extensive experiments demonstrate that UniPart achieves superior segmentation controllability and part-level geometric quality compared with existing approaches.