This paper proposes two consistent one-sided specification tests for parametric regression models, one based on the sample covariance between the residual from the parametric model and the discrepancy between the parametric and nonparametric fitted values; the other based on the difference in sums of squared residuals between the parametric and nonparametric models. We estimate the nonparametric model by series regression. The new test statistics converge in distribution to a unit normal under correct specification and grow to infinity faster than the parametric rate $(n^{-1/2})$ under misspecification, while avoiding weighting, sample splitting, and non-nested testing procedures used elsewhere in the literature. Asymptotically, our tests can be viewed as a test of the joint hypothesis that the true parameters of a series regression model are zero, where the dependent variable is the residual from the parametric model, and the series terms are functions of the explanatory variables, chosen so as to support nonparametric estimation of a conditional expectation. We specifically consider Fourier series and regression splines, and present a Monte Carlo study of the finite sample performance of the new tests in comparison to consistent tests of Bierens (1990), Eubank and Spiegelman (1990), Jayasuriya (1990), Wooldridge (1992), and Yatchew (1992); the results show the new tests have good power, performing quite well in some situations. We suggest a joint Bonferroni procedure that combines a new test with those of Bierens and Wooldridge to capture the best features of the three approaches.