The ongoing crisis of the COVID-19 (Coronavirus disease 2019) pandemic has changed the world. According to the World Health Organization (WHO), 4 million people have died due to this disease, whereas there have been more than 180 million confirmed cases of COVID-19. The collapse of the health system in many countries has demonstrated the need of developing tools to automatize the diagnosis of the disease from medical imaging. Previous studies have used deep learning for this purpose. However, the performance of this alternative highly depends on the size of the dataset employed for training the algorithm. In this work, we propose a classification framework based on sparse coding in order to identify the pneumonia patterns associated with different pathologies. Specifically, each chest X-ray (CXR) image is partitioned into different tiles. The most relevant features extracted from PCA are then used to build the dictionary within the sparse coding procedure. Once images are transformed and reconstructed from the elements of the dictionary, classification is performed from the reconstruction errors of individual patches associated with each image. Performance is evaluated in a real scenario where simultaneously differentiation between four different pathologies: control vs bacterial pneumonia vs viral pneumonia vs COVID-19. The accuracy when identifying the presence of pneumonia is 93.85%, whereas 88.11% is obtained in the 4-class classification context. The excellent results and the pioneering use of sparse coding in this scenario evidence the applicability of this approach as an aid for clinicians in a real-world environment.