For the reliability-oriented sensitivity analysis with respect to the parameters of input variables, by introducing the copula function to describe the joint probability distribution with dependent input variables, the reliability-oriented sensitivity can be decomposed into independent sensitivity and dependent sensitivity, which can be used to measure the influence of distribution parameters separately. Since the parameters of multivariate copula function are difficult to be estimated and not flexible in high dimension, the bivariate copulas are preferred in practice. Then, the vine copula model is employed to transform the multivariate joint probability density function (PDF) into the product of multiple bivariate copulas and marginal PDF of all variables. Based on copula theory, the computation of reliability-oriented sensitivity with dependent variables can be transformed into the computation of a score function for each marginal PDF and the computation of a copula score function for each pair-copula PDF involved in the vine factorization. A general numerical approach is proposed to compute the separate sensitivity. Then, some numerical examples and engineering applications are employed to validate the rationality of the proposed method.