This paper introduces a pioneering experimental study on the automated packing of a catering package using a two-fingered gripper affixed to a 3-degree-of-freedom Delta parallel robot. A distinctive contribution lies in the application of a deep learning approach to tackle this challenge. A custom dataset, comprising 1,500 images, is meticulously curated for this endeavor, representing a noteworthy initiative as the first dataset focusing on Persian-manufactured products. The study employs the YOLOV5 model for object detection, followed by segmentation using the FastSAM model. Subsequently, rotation angle calculation is facilitated with segmentation masks, and a rotated rectangle encapsulating the object is generated. This rectangle forms the basis for calculating two grasp points using a novel geometrical approach involving eigenvectors. An extensive experimental study validates the proposed model, where all pertinent information is seamlessly transmitted to the 3-DOF Delta parallel robot. The proposed algorithm ensures real-time detection, calibration, and the fully autonomous packing process of a catering package, boasting an impressive over 80\% success rate in automatic grasping. This study marks a significant stride in advancing the capabilities of robotic systems for practical applications in packaging automation.