Machine learning and artificial intelligence have shown remarkable performance in accelerated magnetic resonance imaging (MRI). Cloud computing technologies have great advantages in building an easily accessible platform to deploy advanced algorithms. In this work, we develop an open-access, easy-to-use and high-performance medical intelligence cloud computing platform (XCloud-pFISTA) to reconstruct MRI images from undersampled k-space data. Two state-of-the-art approaches of the Projected Fast Iterative Soft-Thresholding Algorithm (pFISTA) family have been successfully implemented on the cloud. This work can be considered as a good example of cloud-based medical image reconstruction and may benefit the future development of integrated reconstruction and online diagnosis system.