Deep learning-based cortical surface reconstruction (CSR) methods heavily rely on pseudo ground truth (pGT) generated by conventional CSR pipelines as supervision, leading to dataset-specific challenges and lengthy training data preparation. We propose a new approach for reconstructing multiple cortical surfaces using weak supervision from brain MRI ribbon segmentations. Our approach initializes a midthickness surface and then deforms it inward and outward to form the inner (white matter) and outer (pial) cortical surfaces, respectively, by jointly learning diffeomorphic flows to align the surfaces with the boundaries of the cortical ribbon segmentation maps. Specifically, a boundary surface loss drives the initialization surface to the target inner and outer boundaries, and an inter-surface normal consistency loss regularizes the pial surface in challenging deep cortical sulci. Additional regularization terms are utilized to enforce surface smoothness and topology. Evaluated on two large-scale brain MRI datasets, our weakly-supervised method achieves comparable or superior CSR accuracy and regularity to existing supervised deep learning alternatives.