Limited access to neurological care leads to missed diagnoses of Parkinson's disease (PD), leaving many individuals unidentified and untreated. We trained a novel neural network-based fusion architecture to detect Parkinson's disease (PD) by analyzing features extracted from webcam recordings of three tasks: finger tapping, facial expression (smiling), and speech (uttering a sentence containing all letters of the alphabet). Additionally, the model incorporated Monte Carlo Dropout to improve prediction accuracy by considering uncertainties. The study participants (n = 845, 272 with PD) were randomly split into three sets: 60% for training, 20% for model selection (hyper-parameter tuning), and 20% for final performance evaluation. The dataset consists of 1102 sessions, each session containing videos of all three tasks. Our proposed model achieved significantly better accuracy, area under the ROC curve (AUROC), and sensitivity at non-inferior specificity compared to any single-task model. Withholding uncertain predictions further boosted the performance, achieving 88.0% (95% CI: 87.7% - 88.4%) accuracy, 93.0% (92.8% - 93.2%) AUROC, 79.3% (78.4% - 80.2%) sensitivity, and 92.6% (92.3% - 92.8%) specificity, at the expense of not being able to predict for 2.3% (2.0% - 2.6%) data. 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. This accessible, low-cost approach requiring only an internet-enabled device with a webcam and microphone paves the way for convenient PD screening at home, particularly in regions with limited access to clinical specialists.