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This paper presents a parallel implementation of the "Interpolated Factored Green Function" (IFGF) method introduced recently for the accelerated evaluation of discrete integral operators arising in wave scattering and other areas (Bauinger and Bruno, Jour. Computat. Phys., 2021). On the basis of the hierarchical IFGF interpolation strategy, the proposed (hybrid MPI-OpenMP) parallel implementation results in highly efficient data communication, and it exhibits in practice excellent parallel scaling up to large numbers of cores-without any hard limitations on the number of cores concurrently employed in an efficient manner. Moreover, on any given number of cores, the proposed parallel approach preserves the O(N log N ) computing cost inherent in the sequential version of the IFGF algorithm. Unlike other approaches, the IFGF method does not utilize the Fast Fourier Transform (FFT), and it is thus better suited than other methods for efficient parallelization in distributed-memory computer systems. A variety of numerical results presented in this paper illustrate the character of the proposed parallel algorithm, including excellent weak and strong parallel scaling properties in all cases considered-for problems of up to 4, 096 wavelengths in electrical size, and scaling tests spanning from 1 compute core to all 1, 680 cores available in the HPC cluster used.