Grasping has been a crucial but challenging problem in robotics for many years. One of the most important challenges is how to make grasping generalizable and robust to novel objects as well as grippers in unstructured environments. We present \regnet, a robotic grasping system that can adapt to different parallel jaws to grasp diversified objects. To support different grippers, \regnet embeds the gripper parameters into point clouds, based on which it predicts suitable grasp configurations. It includes three components: Score Network (SN), Grasp Region Network (GRN), and Refine Network (RN). In the first stage, SN is used to filter suitable points for grasping by grasp confidence scores. In the second stage, based on the selected points, GRN generates a set of grasp proposals. Finally, RN refines the grasp proposals for more accurate and robust predictions. We devise an analytic policy to choose the optimal grasp to be executed from the predicted grasp set. To train \regnet, we construct a large-scale grasp dataset containing collision-free grasp configurations using different parallel-jaw grippers. The experimental results demonstrate that \regnet with the analytic policy achieves the highest success rate of $74.98\%$ in real-world clutter scenes with $20$ objects, significantly outperforming several state-of-the-art methods, including GPD, PointNetGPD, and S4G. The code and dataset are available at https://github.com/zhaobinglei/REGNet-V2.