We present a new solver for massively parallel simulations of fully three-dimensional multiphase flows. The solver runs on a variety of computer architectures from laptops to supercomputers and on 262144 threads or more (limited only by the availability to us of more threads). The code is wholly written by the authors in Fortran 2008 and uses a domain decomposition strategy for parallelization with MPI. The fluid interface solver is based on a parallel implementation of the LCRM hybrid front tracking/level set method designed to handle highly deforming interfaces with complex topology changes. We discuss the implementation of this interface method and its particular suitability to distributed processing where all operations are carried out locally on distributed subdomains. We have developed parallel GMRES and Multigrid iterative solvers suited to the linear systems arising from the implicit solution of the fluid velocities and pressure in the presence of strong density and viscosity discontinuities across fluid phases. Particular attention is drawn to the details and performance of the parallel Multigrid solver. The code includes modules for flow interaction with immersed solid objects, contact line dynamics, species and thermal transport with phase change. Here, however, we focus on the simulation of the canonical problem of drop splash onto a liquid film and report on the parallel performance of the code on varying numbers of threads. The 3D simulations were run on mesh resolutions up to 10243 with results at the higher resolutions showing the fine details and features of droplet ejection, crown formation and rim instability observed under similar experimental conditions.