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Tracking Parkinson's disease (PD) symptom progression often uses the unified Parkinson's disease rating scale (UPDRS) that requires the patient's presence in clinic, and time-consuming physical examinations by trained medical staff. Thus, symptom monitoring is costly and logistically inconvenient for patient and clinical staff alike, also hindering recruitment for future large-scale clinical trials. Here, for the first time, we demonstrate rapid, remote replication of UPDRS assessment with clinically useful accuracy (about 7.5 UPDRS points difference from the clinicians' estimates), using only simple, self-administered, and noninvasive speech tests. We characterize speech with signal processing algorithms, extracting clinically useful features of average PD progression. Subsequently, we select the most parsimonious model with a robust feature selection algorithm, and statistically map the selected subset of features to UPDRS using linear and nonlinear regression techniques that include classical least squares and nonparametric classification and regression trees. We verify our findings on the largest database of PD speech in existence (~6000 recordings from 42 PD patients, recruited to a six-month, multicenter trial). These findings support the feasibility of frequent, remote, and accurate UPDRS tracking. This technology could play a key part in telemonitoring frameworks that enable large-scale clinical trials into novel PD treatments.