This paper addresses the ε-close parameter tuning problem for Bayesian networks (BNs): find a minimal ε-close amendment of probability entries in a given set of (rows in) conditional probability tables that make a given quantitative constraint on the BN valid. Based on the state-of-the-art “region verification” techniques for parametric Markov chains, we propose an algorithm whose capabilities go beyond any existing techniques. Our experiments show that ε-close tuning of large BN benchmarks with up to eight parameters is feasible. In particular, by allowing (i) varied parameters in multiple CPTs and (ii) inter-CPT parameter dependencies, we treat subclasses of parametric BNs that have received scant attention so far.