Objective. Image reconstruction is a fundamental step in magnetic particle imaging (MPI). One of the main challenges is the fact that the reconstructions are computationally intensive and time-consuming, so choosing an algorithm presents a compromise between accuracy and execution time, which depends on the application. This work proposes a method that provides both fast and accurate image reconstructions. Approach. Image reconstruction algorithms were implemented to be executed in parallel in graphics processing units (GPUs) using the CUDA framework. The calculation of the model-based MPI calibration matrix was also implemented in GPU to allow both fast and flexible reconstructions. Main results. The parallel algorithms were able to accelerate the reconstructions by up to about 6,100 times in comparison to the serial Kaczmarz algorithm executed in the CPU, allowing for real-time applications. Reconstructions using the OpenMPIData dataset validated the proposed algorithms and demonstrated that they are able to provide both fast and accurate reconstructions. The calculation of the calibration matrix was accelerated by up to about 37 times. Significance. The parallel algorithms proposed in this work can provide single-frame MPI reconstructions in real time, with frame rates greater than 100 frames per second. The parallel calculation of the calibration matrix can be combined with the parallel reconstruction to deliver images in less time than the serial Kaczmarz reconstruction, potentially eliminating the need of storing the calibration matrix in the main memory, and providing the flexibility of redefining scanning and reconstruction parameters during execution.