Parkinson's disease (PD) is a common neurological disorder, with bradykinesia being one of its cardinal features. Objective quantification of bradykinesia using computer vision has the potential to standardise decision-making, for patient treatment and clinical trials, while facilitating remote assessment. We utilised a dataset of part-3 MDS-UPDRS motor assessments, collected at four independent clinical and one research sites on two continents, to build computer-vision-based models capable of inferring the correct severity rating robustly and consistently across all identifiable subgroups of patients. These results contrast with previous work limited by small sample sizes and small numbers of sites. Our bradykinesia estimation corresponded well with clinician ratings (interclass correlation 0.74). This agreement was consistent across four clinical sites. This result demonstrates how such technology can be successfully deployed into existing clinical workflows, with consumer-grade smartphone or tablet devices, adding minimal equipment cost and time.