In this paper, we describe the application of a new version of Barnett's method to the squarefree decomposition of a univariate polynomial with coefficients in K|x), x being a parameter and K a characteristic zero field. This new version of Barnett's method uses Bezoutian matrices instead of matrices obtained from evaluating polynomials in a companion matrix and allows the determination of the squarefree decomposition parametrizing the god of the polynomial and its successive derivatives with respect to the main variable. The application of this parametric squarefree decomposition to the integration of parametric rational functions is also presented.