This article explores parametric curves defined by polynomials with parametric coefficients. We introduce an automated classification method that utilizes computational algebra techniques and singular variety theory, focusing on various parameters and their singular points—where the derivative is zero or undefined. These singular points are crucial for identifying key curve features. In 2010, Dehghani classified cubic curves using the Montes DISPGB algorithm; in this paper, we apply the Dehghani GESGVWCGS algorithm to classify all families of parametric polynomial curves, including cubic ones. Our results extend previous research, enabling more complex analyses of higher-dimensional varieties and displaying parametric constraints more easily and structurally. We believe that integrating computational tools with singular variety theory will enhance the theory and applications of parametric curve studies in fields like computer graphics and architectural design. This method streamlines curve analysis and improves applicability. All computations are conducted using Maple 18, and the methodology is demonstrated with two families of polynomial curves featuring parametric coefficients.