Motivated by models of biological chemotaxis, we establish the global existence and boundedness of classical solutions for a class of strongly coupled parabolic systems involving cross-diffusion and nonlocal source terms. The key insight is that the pathway to proving global regularity depends critically on the source term's structure, which combines local and nonlocal effects. We will discuss the distinct analytical techniques required for two- and three-component systems, covering both fully parabolic and mixed parabolic-elliptic cases.