Fine-grained 3D part segmentation is crucial for enabling embodied AI systems to perform complex manipulation tasks, such as interacting with specific functional components of an object. However, existing interactive segmentation methods are largely confined to coarse, instance-level targets, while non-interactive approaches struggle with sparse, real-world scans and suffer from a severe lack of annotated data. To address these limitations, we introduce PinPoint3D, a novel interactive framework for fine-grained, multi-granularity 3D segmentation, capable of generating precise part-level masks from only a few user point clicks. A key component of our work is a new 3D data synthesis pipeline that we developed to create a large-scale, scene-level dataset with dense part annotations, overcoming a critical bottleneck that has hindered progress in this field. Through comprehensive experiments and user studies, we demonstrate that our method significantly outperforms existing approaches, achieving an average IoU of around 55.8% on each object part under first-click settings and surpassing 71.3% IoU with only a few additional clicks. Compared to current state-of-the-art baselines, PinPoint3D yields up to a 16% improvement in IoU and precision, highlighting its effectiveness on challenging, sparse point clouds with high efficiency. Our work represents a significant step towards more nuanced and precise machine perception and interaction in complex 3D environments.