The human brain undergoes dynamic, potentially pathology-driven, structural changes throughout a lifespan. Longitudinal Magnetic Resonance Imaging (MRI) and other neuroimaging data are valuable for characterizing trajectories of change associated with typical and atypical aging. However, the analysis of such data is highly challenging given their discrete nature with different spatial and temporal image sampling patterns within individuals and across populations. This leads to computational problems for most traditional deep learning methods that cannot represent the underlying continuous biological process. To address these limitations, we present a new, fully data-driven method for representing aging trajectories across the entire brain by modelling subject-specific longitudinal T1-weighted MRI data as continuous functions using Implicit Neural Representations (INRs). Therefore, we introduce a novel INR architecture capable of partially disentangling spatial and temporal trajectory parameters and design an efficient framework that directly operates on the INRs' parameter space to classify brain aging trajectories. To evaluate our method in a controlled data environment, we develop a biologically grounded trajectory simulation and generate T1-weighted 3D MRI data for 450 healthy and dementia-like subjects at regularly and irregularly sampled timepoints. In the more realistic irregular sampling experiment, our INR-based method achieves 81.3% accuracy for the brain aging trajectory classification task, outperforming a standard deep learning baseline model (73.7%).