The parcellation of the human cortex into meaningful anatomical units is a common step of various neuroimaging studies. There have been multiple successful efforts to process magnetic resonance (MR) brain images automatically and identify specific anatomical regions, following atlases defined from cortical landmarks. Those definitions usually rely first on a high-quality brain surface reconstruction. On the other hand, when high accuracy is not a requirement, simpler methods based on warping a probabilistic atlas have been widely adopted. Here, we develop a cortical parcellation method for MR brain images based on Convolutional Neural Networks (ConvNets), a machine-learning method, with the goal of automatically transferring the knowledge obtained from surface analyses onto something directly applicable on simpler volume data. We train a ConvNet on a large (thousand) set of cortical ribbons of multiple MRI cohorts, to reproduce parcellations obtained from a surface method, in this case FreeSurfer. Further, to make the model applicable in a broader context, we force the model to generalize to unseen segmentations. The model is evaluated on unseen data of unseen cohorts. We characterize the behavior of the model during learning, and quantify its reliance on the dataset itself, which tends to give support for the necessity of large training sets, augmentation, and multiple contrasts. Overall, ConvNets can provide an efficient way to parcel MRI images, following the guidance established within more complex methods, quickly and accurately. The trained model is embedded within a open-source parcellation tool available at https://github.com/bthyreau/parcelcortex.