We describe the parallelization of a computer program for the adaptive mesh refinement simulation of variable density, viscous, incompressible fluid flows for low Mach number combustion. The adaptive methodology is based on the use of local grids superimposed on a coarse grid to achieve sufficient resolution in the solution. The key elements of the approach to parallelization are a dynamic load-balancing technique to distribute work to processors and a software methodology for managing data distribution and communications. The methodology is based on a message-passing model that exploits the coarse-grained parallelism inherent in the algorithms. A method is presented for parallelizing weakly sequential loops—loops with sparse dependencies among iterations.