This paper describes the implementation and performances of a parallel solver for the direct numerical simulation of the three-dimensional and time-dependent Navier -Stokes equations on distributedmemory, massively parallel computers. The feasibility of this approach to study Marangoni flow instability in half zone liquid bridges is examined. The results indicate that the incompressible, non-linear Navier-Stokes problem, governing the Marangoni flows behavior, can effectively be parallelized on a distributed memory parallel machine by remapping the distributed data structure. The numerical code is based on a three-dimensional Simplified Marker and Cell (SMAC) primitive variable method applied to a staggered finite difference grid. Using this method, the problem is split into two problems, one parabolic and the other elliptic A parallel algorithm, explicit in time, is utilized to solve the parabolic equations. A parallel multisplitting kernel is introduced for the solution of the pseudo pressure elliptic equation, representing the most time-consuming part of the algorithm. A grid-partition strategy is used in the parallel implementations of both the parabolic equations and the multisplitting elliptic kernel. A Message Passing Interface (MPI) is coded for the boundary conditions; this protocol is portable to different systems supporting this interface for interprocessor communications. Numerical experiments illustrate good numerical properties and parallel efficiency. In particular, good scalability on a large number of processors can be achieved as long as the granularity of the parallel application is not too small. However, increasing the number of processors, the Speed-Up is ever smaller than the ideal linear Speed-Up. The communication timings indicate that complex practical calculations, such as the solutions of the Navier-Stokes equations for the numerical simulation of the instability of Marangoni flows, can be expected to run on a massively parallel machine with good efficiency.