The article considers parallel strategies and tactics at different stages of mathematical modeling. These technological steps include geometrical and functional modeling, discretization and approximation, algebraic solvers and optimization methods for inverse problems, postprocessing and visualization of numerical results, as well as decision-making systems. Scalable parallelism can be provided by combined application of MPI tools, multi-thread computing, vectorization, and the use of graphics accelerators. The general method to achieve high-performance computing consists in minimizing data communications, which are the most time and energy consuming. The construction of efficient parallel algorithms and code optimization is based on various approaches at different levels of computational schemes. The implementation of the biggest interdisciplinary direct and inverse problems in cloud computing technologies is considered. The corresponding applied software with a long life cycle is represented as integrated environment oriented to large groups of end users.