Eye-tracking is an accessible and non-invasive technology that provides information about a subject's motor and cognitive abilities. As such, it has proven to be a valuable resource in the study of neurodegenerative diseases such as Parkinson's disease. Saccade experiments, in particular, have proven useful in the diagnosis and staging of Parkinson's disease. However, to date, no single eye-movement biomarker has been found to conclusively differentiate patients from healthy controls. In the present work, we investigate the use of state-of-the-art deep learning algorithms to perform Parkinson's disease classification using eye-tracking data from saccade experiments. In contrast to previous work, instead of using hand-crafted features from the saccades, we use raw $\sim1.5\,s$ long fixation intervals recorded during the preparatory phase before each trial. Using these short time series as input we implement two different classification models, InceptionTime and ROCKET. We find that the models are able to learn the classification task and generalize to unseen subjects. InceptionTime achieves $78\%$ accuracy, while ROCKET achieves $88\%$ accuracy. We also employ a novel method for pruning the ROCKET model to improve interpretability and generalizability, achieving an accuracy of $96\%$. Our results suggest that fixation data has low inter-subject variability and potentially carries useful information about brain cognitive and motor conditions, making it suitable for use with machine learning in the discovery of disease-relevant biomarkers.