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This paper considers the problem of testing parametric assumptions on signals f from which only noisy observations yk=f(τk)+εk are available, and where the signal is assumed to be either band-limited or time-limited. To this end, the signal is reconstructed by an estimator based on the Whittaker-Shannon (WS) sampling theorem with oversampling. As test statistic, the minimal L2 distance between the estimated signal and the parametric model is used. To construct appropriate tests, the asymptotic distribution of the test statistic is derived both under the hypothesis of the validity of the parametric model and under fixed local alternatives. As a byproduct, the asymptotic distribution of the integrated square error of the estimator is computed, which is of interest by itself, e.g., for the analysis of a cross-validated bandwidth selector.