Emerging miniaturization technologies for biological and bioengineering applications require precise control over position and actuation of microparticles. While many of these applications call for high-throughput approaches, common tools for particle manipulation, such as magnetic or optical tweezers, suffer from low parallelizability. To address this issue, we introduce a chip-based platform that enables flexible three-dimensional control over individual magnetic microparticles. Our system relies on microwire crossbar arrays for simultaneous generation of magnetic and dielectric forces, which actuate the particles along highly localized traps. We demonstrate the precise spatiotemporal control of individual particles by tracing complex trajectories in three dimensions and investigate the forces that can be generated along different axes. Furthermore, we show that our approach for particle actuation can be parallelized by simultaneously controlling the position and movement of 16 particles in parallel.