This paper proposes to encode the distribution of features learned from a convolutional neural network using a Gaussian Mixture Model. These parametric features, called GMM-CNN, are derived from chest computed tomography and X-ray scans of patients with Coronavirus Disease 2019. We use the proposed GMM-CNN features as input to a robust classifier based on random forests to differentiate between COVID-19 and other pneumonia cases. Our experiments assess the advantage of GMM-CNN features compared to standard CNN classification on test images. Using a random forest classifier (80\% samples for training; 20\% samples for testing), GMM-CNN features encoded with two mixture components provided a significantly better performance than standard CNN classification (p\,$<$\,0.05). Specifically, our method achieved an accuracy in the range of 96.00\,--\,96.70\% and an area under the ROC curve in the range of 99.29\,--\,99.45\%, with the best performance obtained by combining GMM-CNN features from both computed tomography and X-ray images. Our results suggest that the proposed GMM-CNN features could improve the prediction of COVID-19 in chest computed tomography and X-ray scans.