In this paper, we present a novel deep neural network architecture for joint class-agnostic object segmentation and grasp detection for robotic picking tasks using a parallel-plate gripper. We introduce depth-aware Coordinate Convolution (CoordConv), a method to increase accuracy for point proposal based object instance segmentation in complex scenes without adding any additional network parameters or computation complexity. Depth-aware CoordConv uses depth data to extract prior information about the location of an object to achieve highly accurate object instance segmentation. These resulting segmentation masks, combined with predicted grasp candidates, lead to a complete scene description for grasping using a parallel-plate gripper. We evaluate the accuracy of grasp detection and instance segmentation on challenging robotic picking datasets, namely Sil\'eane and OCID_grasp, and show the benefit of joint grasp detection and segmentation on a real-world robotic picking task.