This paper proposes a deep learning-based fast grasp detection method with a small dataset for robotic bin-picking. We consider the problem of grasping stacked up mechanical parts on a planar workspace using a parallel gripper. In this paper, we use a deep neural network to solve the problem with a single depth image. To reduce the computation time, we propose an edge-based algorithm to generate potential grasps. Then, a convolutional neural network (CNN) is applied to evaluate the robustness of all potential grasps for bin-picking. Finally, the proposed method ranks them and the object is grasped by using the grasp with the highest score. In bin-picking experiments, we evaluate the proposed method with a 7-DOF manipulator using textureless mechanical parts with complex shapes. The success ratio of grasping is 97%, and the average computation time of CNN inference is less than 0.23[s] on a laptop PC without a GPU array. In addition, we also confirm that the proposed method can be applied to unseen objects which are not included in the training dataset.