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Face parsing assigns pixel-wise semantic labels as the face representation for computers, which is the fundamental part of many advanced face technologies. Compared with 2D face parsing, 3D face parsing shows more potential to achieve better performance and further application, but it is still challenging due to 3D mesh data computation. Recent works introduced different methods for 3D surface segmentation, while the performance is still limited. In this paper, we propose a method based on the "3D-2D-3D" strategy to accomplish 3D face parsing. The topological disk-like 2D face image containing spatial and textural information is transformed from the sampled 3D face data through the face parameterization algorithm, and a specific 2D network called CPFNet is proposed to achieve the semantic segmentation of the 2D parameterized face data with multi-scale technologies and feature aggregation. The 2D semantic result is then inversely re-mapped to 3D face data, which finally achieves the 3D face parsing. Experimental results show that both CPFNet and the "3D-2D-3D" strategy accomplish high-quality 3D face parsing and outperform state-of-the-art 2D networks as well as 3D methods in both qualitative and quantitative comparisons.