Eye-tracking is an accessible, non-invasive technology that provides information on motor and cognitive function and is therefore relevant for studying neurodegenerative diseases. Saccade experiments, in particular, have proven useful in the study of Parkinson's disease progression. However, to date, no single eye-movement biomarker has been shown to reliably differentiate patients from healthy controls. In the present work, we investigate state-of-the-art deep learning methods for Parkinson's disease classification using eye-tracking time-series data from saccade experiments. We analyzed eye-tracking data recorded from 84 participants (54 patients with Parkinson's disease and 30 healthy controls) during prosaccade and antisaccade tasks. Unlike most prior work, we do not rely on hand-crafted features derived from saccades to train the classification algorithms. Instead, we use raw ∼1.5 s fixation intervals extracted from the preparatory phase of the task. We compare three time-series classifiers: InceptionTime, ROCKET, and Detach-ROCKET (a pruned ROCKET variant). On subject-level classification of unseen individuals, the models achieve 78%, 92%, and 95% accuracy, respectively. We further examine interpretability of the best-performing model by relating its confidence scores to patient metadata and by analyzing the features retained after pruning. Our findings indicate that fixation data contains disease-relevant information and enables classification that is robust to subject-specific fingerprints, supporting its use for machine-learning-driven biomarker discovery in neurodegenerative diseases.