The impact of incomplete surface accommodation on non-equilibrium heat transfer in a lid-driven cavity is evaluated by a parallel direct simulation Monte Carlo method. A two-dimensional partitioning technique has been employed to parallelize the code and its parallel performance is assessed. The computed results indicate that incomplete surface accommodation significantly impacts all aspects of flow and heat transfer in the cavity like the vortex center, wall heat flux rates, and the heat transfer mechanism itself. In particular, a counter-gradient heat flux pattern exists when the flow is in a non-equilibrium state, under which thermal energy is transferred from a cold region to a hot region. However, the gaseous heat transfer process changes with any increase in incomplete surface accommodation, significantly affecting the direction of heat transfer.