In this paper we consider the problem of constant-time evaluation of subdivision surfaces at arbitrary points. Our work extends the work of J. Stam, by considering the subdivision rules for piecewise smooth surfaces with boundaries depending on parameters. The main innovation of this paper is the idea of using a different set of basis vectors for evaluation, which, unlike eigenvectors, depend continuously on the coefficients of the subdivision rules. The advantage of this approach is that it becomes possible to define evaluation for parametric families of rules without considering excessive number of special cases, while improving numerical stability of calculations. We demonstrate how such bases are computed for a particular parametric family of subdivision rules extending Loop subdivision to meshes with boundary, and provide a detailed description of the evaluation algorithms.