Parkinson's disease (PD) is the second most common neurodegenerative disorder, mainly characterized by motor alterations. Despite multiple efforts, there is no definitive biomarker to diagnose, quantify, and characterize the disease early. Recently, abnormal fixational oculomotor patterns have emerged as a promising disease biomarker with high sensitivity, even at early stages. Nonetheless, the complex patterns and potential correlations with the disease remain largely unexplored, among others, because of the limitations of standard setups that only analyze coarse measures and poorly exploit the associated PD alterations. This work introduces a new strategy to represent, analyze and characterize fixational patterns from non-invasive video analysis, adjusting a geometric learning strategy. A deep Riemannian framework is proposed to discover potential oculomotor patterns aimed at withstanding data scarcity and geometrically interpreting the latent space. A convolutional representation is first built, then aggregated onto a symmetric positive definite matrix (SPD). The latter encodes second-order statistics of deep convolutional features and feeds a non-linear hierarchical architecture that processes SPD data by maintaining them into their Riemannian manifold. The complete representation discriminates between Parkinson and Healthy (Control) fixational observations, even at PD stages 2.5 and 3. Besides, the proposed geometrical representation exhibit capabilities to statistically differentiate observations among Parkinson's stages. The developed tool demonstrates coherent results from explainability maps back-propagated from output probabilities.