Partial differential equations (PDEs) are crucial in modeling diverse phenomena across scientific disciplines, including seismic and medical imaging, computational fluid dynamics, image processing, and neural networks. Solving these PDEs at scale is an intricate and time-intensive process that demands careful tuning. This paper introduces automated code-generation techniques specifically tailored for distributed memory parallelism (DMP) to execute explicit finite-difference (FD) stencils at scale, a fundamental challenge in numerous scientific applications. These techniques are implemented and integrated into the Devito DSL and compiler framework, a well-established solution for automating the generation of FD solvers based on a high-level symbolic math input. Users benefit from modeling simulations for real-world applications at a high-level symbolic abstraction and effortlessly harnessing HPC-ready distributed-memory parallelism without altering their source code. This results in drastic reductions both in execution time and developer effort. A comprehensive performance evaluation of Devito's DMP via MPI demonstrates highly competitive strong and weak scaling on CPU and GPU clusters, proving its effectiveness and capability to meet the demands of large-scale scientific simulations.