The C³ parametric interpolation spline function is presented this paper, which has the similar properties of the classical cubic Hermite interpolation spline with additional flexibility and high approximation rates. Moreover, a group of eighth-degree bases with three parameters is constructed. Then, the interpolation spline function is defined based on the proposed basis functions. And the interpolation error and the technique for determining the optimal interpolation are also given. The results show that when the interpolation conditions remain unchanged, the proposed interpolation spline functions retain C³ continuity, and the shape of the curve can be controlled by the parameters. When the optimal values of parameters are chosen, the interpolation spline function can achieve higher approximation rates.