In this paper a method is proposed to maximize the bandwidth of the WDM router based on one-pump fiber parametric wavelength converters. It is proved that for such converters there exists an optimum signal (idler) frequency at which the output (input) tuning range can be maximized. Analytical expressions of the optimum frequency and the maximal tuning range are deduced. Then a two-stage bidirectional wavelength conversion method is proposed. With this method the bandwidth of the WDM router based on such a converter can be significantly improved compared to the one-stage ones by 252% if ordinary highly nonlinear fibers are used or 390% if fibers with optimal fourth order dispersion are used.