The composite likelihood (CL) is amongst the computational methods used for the estimation of high-dimensional multivariate normal (MVN) copula models with discrete responses. Its computational advantage, as a surrogate likelihood method, is that is based on the independence likelihood for the univariate marginal regression and non-regression parameters and pairwise likelihood for the correlation parameters. Nevertheless, the efficiency of the CL method for estimating the univariate regression and non-regression marginal parameters can be low. For a high-dimensional discrete response, weighted versions of the composite likelihood estimating equations and an iterative approach to determine good weight matrices are proposed. The general methodology is applied to the MVN copula with univariate ordinal regressions as the marginals. Efficiency calculations show that the proposed method is nearly as efficient as the maximum likelihood for fully specified MVN copula models. Illustrations include simulations and real data applications regarding longitudinal (low-dimensional) and time (high-dimensional) series ordinal response data with covariates. Our studies suggest that there is a substantial gain in efficiency via the weighted CL method.