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Probabilistic voltage stability assessment (VSA) of distribution systems has recently gained significant research attention due to the proliferation of renewable energy sources (RESs). This study presents a nonparametric data-driven VSA in a look-ahead framework by considering correlated variables. First, an accurate but relatively fast correlation model is developed by considering stochastic interdependency with unknown dependency structure. Then, the probabilistic density function (PDF) of the voltage stability index is estimated for near-future minutes. The proposed method employs Bernstein vine copula (BVC) for estimating the unknown underlying dependence structure among different variables. This non-parametric method imposes no assumption on the dependence structure and can handle the problems where no information is available for judging about dependence structure and its belongingness to a specific type of copula function. Additionally, global sensitivity analysis (GSA), with an embedded surrogate model based on the extreme learning machine (ELM), is utilized to rank the loads and RESs according to their contribution to voltage stability margin. Furthermore, a parallel implementation of vine copula construction and dimension reduction is developed based on the GSA to speed up the proposed VSA. The efficiency of the proposed method is examined by simulating the results of modified IEEE 33-node and 123-node systems.