We explore how the big-three computing paradigms---symmetric multiprocessor, graphical processing units (GPUs), and cluster computing---can together be brought to bear on large-data Gaussian processes (GP) regression problems via a careful implementation of a newly developed local approximation scheme. Our methodological contribution focuses primarily on GPU computation, as this requires the most care and also provides the largest performance boost. However, in our empirical work we study the relative merits of all three paradigms to determine how best to combine them. The paper concludes with two case studies. One is a real data fluid-dynamics computer experiment which benefits from the local nature of our approximation; the second is a synthetic example designed to find the largest data set for which (accurate) GP emulation can be performed on a commensurate predictive set in under an hour.