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This work introduces the concept of parametric Gaussian processes (PGP), which is built upon the seemingly self-contradictory idea of making Gaussian processes parametric. The resulting framework is capable of encoding massive amount of data into a small number of "hypothetical" data points. Moreover, parametric Gaussian processes are well aware of their imperfections and are capable of properly quantifying the uncertainty in their predictions associated with such limitations. The effectiveness of the proposed approach is demonstrated using three illustrative examples, including one with simulated data, a benchmark with dataset in the airline industry with approximately 6 million records, and spatio-temporal sea surface temperature maps in Massachusetts and Cape Cod Bays and Stellwagen Bank for the year 2015. [ABSTRACT FROM AUTHOR]