A new parallel implicit solver for the solution of the compressible Navier-Stokes equations with finite rate chemistry on unstructured finite volume meshes is presented. The solver employs the Data-Parallel Line Relaxation (DPLR) method for implicit time integration along lines of cells that are normal to the wall. In regions where surface-normal lines cannot be formed, a point-implicit method is used. The new method combines the robustness and efficiency of the implicit DPLR method with the flexibility of unstructured mesh generation. The solver employs low-dissipation numerical schemes common among structured solvers, as well as a MUSCL-type scheme designed for unstructured discretizations. Partitioning and load balancing of the computational mesh is automatic, and speed-up tests show that the code scales very well on commodity hardware. Simulations show that results obtained with the present solver are comparable to results obtained with the NASA Ames DPLR code. Simulations of the flow through an inward-turning scramjet inlet are presented.