Optic disc (OD) and optic cup (OC) are regions of prominent clinical interest in a retinal fundus image. They are the primary indicators of a glaucomatous condition. With the advent and success of deep learning for healthcare research, several approaches have been proposed for the segmentation of important features in retinal fundus images. We propose a novel approach for the simultaneous segmentation of the OD and OC using a residual encoder-decoder network (REDNet) based regional convolutional neural network (RCNN). The RED-RCNN is motivated by the Mask RCNN (MRCNN). Performance comparisons with the state-of-the-art techniques and extensive validations on standard publicly available fundus image datasets show that RED-RCNN has superior performance compared with MRCNN. RED-RCNN results in Sensitivity, Specificity, Accuracy, Precision, Dice and Jaccard indices of 95.64%, 99.9%, 99.82%, 95.68%, 95.64%, 91.65%, respectively, for OD segmentation, and 91.44%, 99.87%, 99.83%, 85.67%, 87.48%, 78.09%, respectively, for OC segmentation. Further, we perform two-stage glaucoma severity grading using the cup-to-disc ratio (CDR) computed based on the obtained OD/OC segmentation. The superior segmentation performance of RED-RCNN over MRCNN translates to higher accuracy in glaucoma severity grading.