A large number of surface-based analyses on brain imaging data adopt some specific brain atlases to better assess structural and functional changes in one or more brain regions. In these analyses, it is necessary to obtain an anatomically correct surface parcellation scheme in an individual brain by referring to the given atlas. Traditional ways to accomplish this goal are through a designed surface-based registration or hand-crafted surface features, although both of them are time-consuming. A recent deep learning approach depends on a regular spherical parameterization of the mesh, which is computationally prohibitive in some cases and may also demand further post-processing to refine the network output. Therefore, an accurate and fully-automatic cortical surface parcellation scheme directly working on the original brain surfaces would be highly advantageous. In this study, we propose an end-to-end deep brain cortical parcellation network, called DBPN. Through intrinsic and extrinsic graph convolution kernels, DBPN dynamically deciphers neighborhood graph topology around each vertex and encodes the deciphered knowledge into node features. Eventually, a non-linear mapping between the node features and parcellation labels is constructed. Our model is a two-stage deep network which contains a coarse parcellation network with a U-shape structure and a refinement network to fine-tune the coarse results. We evaluate our model in a large public dataset and our work achieves superior performance than state-of-the-art baseline methods in both accuracy and efficiency.