In this work, we propose a procedure to compute Pareto-optimal fronts for the bi-objective Minimum Diameter-Cost Spanning Tree problem (bi-MDCST). The bi-MDCST aims at finding spanning trees with minimum total cost and minimum diameter. Strategic decision problems for high-speed trains infrastructure, as well as tactical and operational optimization problems for network design and transportation can be modeled as bi-MDCST. The proposed exact procedure makes use of components from the multi-objective exact method Parallel Partitioning Method, and Pareto-optimal fronts have been computed for two benchmark instances from the literature. To the best of our knowledge, there are no works dedicated to providing Pareto-optimal fronts for the bi-MDCST.