This study delves into the realms of Parametric Design and Auxetic Structures, focusing on structures exhibiting unconventional behavior, i.e., those that have a mechanical function inverse to the conventional one and expand when subjected to an external force (giving them anti-rubber properties due to their negative Poisson's ratio). It explores the theoretical underpinnings, historical evolution, and diverse applications of these structures across architecture, engineering, biology, design, and art. Through the lens of three case studies, the study showcases the utilization of Rhinoceros and Grasshopper software used for designing grids based on auxetic structures, offering versatility in adapting to various shapes and dimensions. Furthermore, the authors introduce a parametric algorithm leveraging Rhinoceros and Grasshopper digital tools, facilitating the manipulation of tessellations' dimensions, quantity, and line thickness. This algorithm generates intricate three-dimensional models amenable to 3D printing technology. The research concludes with an insightful analysis of the potential applications of these technologies, emphasizing their inherent advantages and the challenges they pose for design and innovation across multiple domains of knowledge. By shedding light on the transformative capabilities of parametric design and auxetic structures, this work underscores their significance in fostering innovation and pushing the boundaries of traditional design paradigms.