Real-world datasets often exhibit imbalanced data distribution, where certain class levels are severely underrepresented. In such cases, traditional pattern classifiers have shown a bias towards the majority class, impeding accurate predictions for the minority class. This paper introduces an imbalanced data-oriented classifier using probabilistic neural networks (PNN) with a skew-normal kernel function to address this major challenge. PNN is known for providing probabilistic outputs, enabling quantification of prediction confidence, interpretability, and the ability to handle limited data. By leveraging the skew-normal distribution, which offers increased flexibility, particularly for imbalanced and non-symmetric data, our proposed Skew-Probabilistic Neural Networks (SkewPNN) can better represent underlying class densities. Hyperparameter fine-tuning is imperative to optimize the performance of the proposed approach on imbalanced datasets. To this end, we employ a population-based heuristic algorithm, the Bat optimization algorithm, to explore the hyperparameter space effectively. We also prove the statistical consistency of the density estimates, suggesting that the true distribution will be approached smoothly as the sample size increases. Theoretical analysis of the computational complexity of the proposed SkewPNN and BA-SkewPNN is also provided. Numerical simulations have been conducted on different synthetic datasets, comparing various benchmark-imbalanced learners. Real-data analysis on several datasets shows that SkewPNN and BA-SkewPNN substantially outperform most state-of-the-art machine-learning methods for both balanced and imbalanced datasets (binary and multi-class categories) in most experimental settings.