Screening for the diagnosis of glaucoma through a fundus image can be determined by the optic cup to disc diameter ratio (CDR), which requires the segmentation of the cup and disc regions. In this paper, we propose two novel approaches, namely Parameter-Shared Branched Network (PSBN) andWeak Region of Interest Model-based segmentation (WRoIM) to identify disc and cup boundaries. Unlike the previous approaches, the proposed methods are trained end-to-end through a single neural network architecture and use dynamic cropping instead of manual or traditional computer vision-based cropping. We are able to achieve similar performance as that of state-of-the-art approaches with less number of network parameters. Our experiments include comparison with different best known methods on publicly available Drishti-GS1 and RIM-ONE v3 datasets. With $7.8 \times 10^6$ parameters our approach achieves a Dice score of 0.96/0.89 for disc/cup segmentation on Drishti-GS1 data whereas the existing state-of-the-art approach uses $19.8\times 10^6$ parameters to achieve a dice score of 0.97/0.89.