Image moments are image descriptors widely utilized in several image processing, pattern recognition, computer vision, and multimedia security applications. In the era of big data, the computation of image moments yields a huge memory demand, especially for large moment order and/or high-resolution images (i.e., megapixel images). The state-of-the-art moment computation methods successfully accelerate the image moment computation for digital images of a resolution smaller than 1K × 1K pixels. For digital images of higher resolutions, image moment computation is problematic. Researchers utilized GPU-based parallel processing to overcome this problem. In practice, the parallel computation of image moments using GPUs encounters the non-extended memory problem, which is the main challenge. This paper proposed a recurrent-based method for computing the Polar Complex Exponent Transform (PCET) moments of fractional orders. The proposed method utilized the symmetry of the image kernel to reduce kernel computation. In the proposed method, once a kernel value is computed in one quaternion, the other three corresponding values in the remaining three quaternions can be trivially computed. Moreover, the proposed method utilized recurrence equations to compute kernels. Thus, the required memory to store the pre-computed memory is saved. Finally, we implemented the proposed method on the GPU parallel architecture. The proposed method overcomes the memory limit due to saving the kernel's memory. The experiments show that the proposed parallel-friendly and memory-efficient method is superior to the state-of-the-art moment computation methods in memory consumption and runtimes. The proposed method computes the PCET moment of order 50 for an image of size 2K × 2K pixels in 3.5 seconds while the state-of-the-art method of comparison needs 7.0 seconds to process the same image, the memory requirements for the proposed method and the method of comparison for the were 67.0 MB and 3.4 GB, respectively. The method of comparison could not compute the image moment for any image with a resolution higher than 2K × 2K pixels. In contrast, the proposed method managed to compute the image moment up to 16K × 16K pixels image.