Legendre moments and their invariants for 2D and 3D image/objects are widely used in image processing, computer vision, and pattern recognition applications. Reconstruction of digital images by nature required higher-order moments to get high-quality reconstructed images. Different applications such as classification of bacterial contamination images utilize high-order moments for feature extraction phase. For big size images and 3D objects, Legendre moments computation is very time-consuming and compute-intensive. This problem limits the use of Legendre moments and makes them impractical for real-time applications. Multi-core CPUs and GPUs are powerful processing parallel architectures. In this paper, new parallel algorithms are proposed to speed up the process of exact Legendre moments computation for 2D and 3D image/objects. These algorithms utilize multi-core CPUs and GPUs parallel architectures where each pixel/voxel of the input digital image/object can be handled independently. A detailed profile analysis is presented where the weight of each part of the entire computational process is evaluated. In addition, we contributed to the parallel 2D/3D Legendre moments by: (1) a modification of the traditional exact Legendre moment algorithm to better fit the parallel architectures, (2) we present the first parallel CPU implementation of Legendre moment, and (3) we present the first parallel CPU and GPU acceleration of the reconstruction phase of the Legendre moments. A set of numerical experiments with different gray-level images are performed. The obtained results clearly show a very close to optimal parallel gain. The extreme reduction in execution times, especially for 8-core CPUs and GPUs, makes the parallel exact 2D/3D Legendre moments suitable for real-time applications.