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We consider parallel computation for Gaussian process calculations to overcome computational and memory constraints on the size of datasets that can be analyzed. Using a hybrid parallelization approach that uses both threading (shared memory) and message-passing (distributed memory), we implement the core linear algebra operations used in spatial statistics and Gaussian process regression in an R package called bigGP that relies on C and MPI. The approach divides the covariance matrix into blocks such that the computational load is balanced across processes while communication between processes is limited. The package provides an API enabling R programmers to implement Gaussian process-based methods by using the distributed linear algebra operations without any C or MPI coding. We illustrate the approach and software by analyzing an astrophysics dataset with n = 67, 275 observations.