Parking availability prediction is a critical component of intelligent transportation systems, aiming to reduce congestion and improve urban mobility. While traditional deep learning models such as Long Short-Term Memory (LSTM) networks have been widely applied, they lack mechanisms to quantify uncertainty, limiting their robustness in real-world deployments. This paper proposes a Bayesian Neural Network (BNN)-based framework for parking occupancy prediction that explicitly models both epistemic and aleatoric uncertainty. Although BNNs have shown promise in other domains, they remain underutilised in parking prediction—likely due to the computational complexity and the absence of real-time context integration in earlier approaches. Our approach leverages contextual features, including temporal and environmental factors, to enhance uncertainty-aware predictions. The framework is evaluated under varying data conditions, including data scarcity (90%, 50%, and 10% of training data) and synthetic noise injection to simulate aleatoric uncertainty. Results demonstrate that BNNs outperform other methods, achieving an average accuracy improvement of 27.4% in baseline conditions, with consistent gains under limited and noisy data. Applying uncertainty thresholds at 20% and 30% further improves reliability by enabling selective, confidence-based decision making. This research shows that modelling both types of uncertainty leads to significantly improved predictive performance in intelligent transportation systems and highlights the potential of uncertainty-aware approaches as a foundation for future work on integrating BNNs with hybrid neuro-symbolic reasoning to enhance decision making under uncertainty.