An accurate low-energy dispersive parametrization of the scalar Kpi form factor was constructed some time ago in terms of a single parameter guided by the Callan-Treiman low-energy theorem. A similar twice-subtracted dispersive parametrization for the vector Kpi form factor will be investigated here. The robustness of the parametrization of these two form factors will be studied in great detail. In particular the cutoff dependence, the isospin breaking effects, and the possible, though not highly probable, presence of zeros in the form factors will be discussed. Interesting constraints in the latter case will be obtained from the soft-kaon analog of the Callan-Treiman theorem and a comparison with the recent tau-->Kpinutau data.