A parallel adaptive mesh refinement (AMR) algorithm is proposed for predicting turbulent non-premixed combusting flows characteristic of gas turbine engine combustors. The Favre-averaged Navier-Stokes equations governing mixture and species transport for a reactive mixture of thermally perfect gases in two dimensions, the two transport equations of the k-ω turbulence model, and the time-averaged species transport equations, are all solved using a fully coupled finite-volume formulation. A flexible block-based hierarchical data structure is used to maintain the connectivity of the solution blocks in the multi-block mesh and facilitate automatic solution-directed mesh adaptation according to physics-based refinement criteria. This AMR approach allows for anisotropic mesh refinement and the block-based data structure readily permits efficient and scalable implementations of the algorithm on multi-processor architectures. Numerical results for turbulent non-premixed diffusion flames, including coldand hot-flow predictions for a bluff body burner, are described and compared to available experimental data. The numerical results demonstrate the validity and potential of the parallel AMR approach for predicting complex non-premixed turbulent combusting flows.