Musculoskeletal diseases and cognitive impairments in patients lead to difficulties in movement as well as negative effects on their psychological health. Clinical gait analysis, a vital tool for early diagnosis and treatment, traditionally relies on expensive optical motion capture systems. Recent advances in computer vision and deep learning have opened the door to more accessible and cost-effective alternatives. This paper introduces a novel spatio-temporal Transformer network to estimate critical gait parameters from RGB videos captured by a single-view camera. Empirical evaluations on a public dataset of cerebral palsy patients indicate that the proposed framework surpasses current state-of-the-art approaches and show significant improvements in predicting general gait parameters (including Walking Speed, Gait Deviation Index - GDI, and Knee Flexion Angle at Maximum Extension), while utilizing fewer parameters and alleviating the need for manual feature extraction.