In this paper, we propose an efficient parallelization strategy for boundary element method (BEM) solvers that perform the electromagnetic analysis of structures with lossy conductors. The proposed solver is accelerated with the adaptive integral method, can model both homogeneous and multilayered background media, and supports excitation via lumped ports or an incident field. Unlike existing parallel BEM solvers, we use a formulation that rigorously models the skin effect, which results in two coupled computational workloads. The external-problem workload models electromagnetic coupling between conductive objects, while the internal-problem workload describes field distributions within them. We propose a parallelization strategy that distributes these two workloads evenly over thousands of processing cores. The external-problem workload is balanced in the same manner as existing parallel solvers that employ approximate models for conductive objects. However, we assert that the internal-problem workload should be balanced by algorithms from scheduling theory. The parallel scalability of the proposed solver is tested on three different structures found in both integrated circuits and metasurfaces. The proposed parallelization strategy runs efficiently on distributed-memory computers with thousands of CPU cores and outperforms competing strategies derived from existing methods.