<p>Model adequacy and heteroscedasticity are key topics in research, aimed at developing methods to prevent model mis-specification in parametric regression models. In this work, we consider two types of hypothesis testing problems of the models: the adequacy test and heteroscedasticity test. Most existing test methods often struggle with the curse of dimensionality, and the limiting null distribution of the test statistic is not asymptotically distribution-free. To address these challenges, a double resampling Kolmogorov-Smirnov test approach is proposed with a projection-based residual-marked empirical process. The asymptotic properties of the test statistic are rigorously investigated under both the null hypothesis and alternative hypotheses. Furthermore, we construct a wild bootstrap procedure for the adequacy test and a residual-based bootstrap procedure for the heteroscedasticity test. The proposed methods are validated to have both the advantages of dimension reduction and the validity of the approximation of the bootstrap, simultaneously. Extensive simulation studies and an analysis of real data are conducted to demonstrate the performance of the proposed method.</p>