This paper deals with a situation when one is interested in the dependence structure of a multidimensional response variable in the presence of a multivariate covariate. It is assumed that the covariate affects only the marginal distributions through regression models while the dependence structure, which is described by a copula, is unaffected. A parametric estimation of the copula function is considered with focus on the maximum pseudo-likelihood method. It is proved that under some appropriate regularity assumptions the estimator calculated from the residuals is asymptotically equivalent to the estimator based on the unobserved errors. In such case one can ignore the fact that the response is first adjusted for the effect of the covariate. A Monte Carlo simulation study explores (among others) situations where the regularity assumptions are not satisfied and the claimed result does not hold. It shows that in such situations the maximum pseudo-likelihood estimator may behave poorly and the moment estimation of the copula parameter is of interest. Our results complement the results available for nonparametric estimation of the copula function.