Estimating a child's age from ocular biometric images is challenging due to subtle physiological changes and the limited availability of longitudinal datasets. Although most biometric age estimation studies have focused on facial features and adult subjects, pediatric-specific analysis, particularly of the iris and periocular regions, remains relatively unexplored. This study presents a comparative evaluation of iris and periocular images for estimating the ages of children aged between 4 and 16 years. We utilized a longitudinal dataset comprising more than 21,000 near-infrared (NIR) images, collected from 288 pediatric subjects over eight years using two different imaging sensors. A multi-task deep learning framework was employed to jointly perform age prediction and age-group classification, enabling a systematic exploration of how different convolutional neural network (CNN) architectures, particularly those adapted for non-square ocular inputs, capture the complex variability inherent in pediatric eye images. The results show that periocular models consistently outperform iris-based models, achieving a mean absolute error (MAE) of 1.33 years and an age-group classification accuracy of 83.82%. These results mark the first demonstration that reliable age estimation is feasible from children's ocular images, enabling privacy-preserving age checks in child-centric applications. This work establishes the first longitudinal benchmark for pediatric ocular age estimation, providing a foundation for designing robust, child-focused biometric systems. The developed models proved resilient across different imaging sensors, confirming their potential for real-world deployment. They also achieved inference speeds of less than 10 milliseconds per image on resource-constrained VR headsets, demonstrating their suitability for real-time applications.