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Dimension-adaptive sparse grid interpolation is a powerful tool to obtain surrogate functions of smooth, medium to high-dimensional objective models. In case of expensive models, the efficiency of the sparse grid algorithm is governed by the time required for the function evaluations. In this paper, we first briefly analyze the inherent parallelism of the standard dimension-adaptive algorithm. Then, we present an enhanced version of the standard algorithm that permits, in each step of the algorithm, a specified number (equal to the number of desired processes) of function evaluations to be executed in parallel, thereby increasing the parallel efficiency.