FluSI: a novel parallel simulation tool for flapping insect flight using a Fourier method with volume penalization. We introduce FluSI, a fully parallel open source software package for pseudospectral simulations of three-dimensional flapping flight in viscous flows. It is freely available for noncommercial use from GitHub (url{https://github.com/pseudospectators/FLUSI}). The computational framework runs on high performance computers with distributed memory architectures. The discretization of the three-dimensional incompressible Navier-Stokes equations is based on a Fourier pseudospectral method with adaptive time stepping. The complex time varying geometry of insects with rigid flapping wings is handled using the volume penalization method. The modules characterizing the insect geometry, flight mechanics, and wing kinematics are described. Validation tests for different benchmarks illustrate the efficiency and precision of the approach. Finally, computations for a model insect in the turbulent regime demonstrate the versatility of the software.