Limited accessibility to neurological care leads to underdiagnosed Parkinson's Disease (PD), preventing early intervention. Existing AI-based PD detection methods primarily focus on unimodal analysis of motor or speech tasks, overlooking the multifaceted nature of the disease. To address this, we introduce a large-scale, multi-task video dataset consisting of $1102$ sessions (each containing videos of finger tapping, facial expression, and speech tasks captured via webcam) from $845$ participants ($272$ with PD). We propose a novel Uncertainty-calibrated Fusion Network (UFNet) that leverages this multimodal data to enhance diagnostic accuracy. UFNet employs independent task-specific networks, trained with Monte Carlo Dropout for uncertainty quantification, followed by self-attended fusion of features, with attention weights dynamically adjusted based on task-specific uncertainties. To ensure patient-centered evaluation, the participants were randomly split into three sets: 60% for training, 20% for model selection, and 20% for final performance evaluation. UFNet significantly outperformed single-task models in terms of accuracy, area under the ROC curve (AUROC), and sensitivity while maintaining non-inferior specificity. Withholding uncertain predictions further boosted the performance, achieving $88.0 \pm 0.3$% accuracy, $93.0 \pm 0.2$% AUROC, $79.3 \pm 0.9$% sensitivity, and $92.6 \pm 0.3$% specificity, at the expense of not being able to predict for $2.3 \pm 0.3$% data ($\pm$ denotes 95% confidence interval). Further analysis suggests that the trained model does not exhibit any detectable bias across sex and ethnic subgroups and is most effective for individuals aged between 50 and 80. Requiring only a webcam and microphone, our approach facilitates accessible home-based PD screening, especially in regions with limited healthcare resources.