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The reconstruction of cerebral cortex surfaces from brain MRI scans is instrumental for the analysis of brain morphology and the detection of cortical thinning in neurodegenerative diseases like Alzheimer's disease (AD). Moreover, for a fine-grained analysis of atrophy patterns, the parcellation of the cortical surfaces into individual brain regions is required. For the former task, powerful deep learning approaches, which provide highly accurate brain surfaces of tissue boundaries from input MRI scans in seconds, have recently been proposed. However, these methods do not come with the ability to provide a parcellation of the reconstructed surfaces. Instead, separate brain-parcellation methods have been developed, which typically consider the cortical surfaces as given, often computed beforehand with FreeSurfer. In this work, we propose two options, one based on a graph classification branch and another based on a novel generic 3D reconstruction loss, to augment template-deformation algorithms such that the surface meshes directly come with an atlas-based brain parcellation. By combining both options with two of the latest cortical surface reconstruction algorithms, we attain highly accurate parcellations with a Dice score of 90.2 (graph classification branch) and 90.4 (novel reconstruction loss) together with state-of-the-art surfaces.