Telemonitoring is the use of electronic devices including smartphones to remotely monitor patients. Using predictive models, telemonitoring data can be translated into a clinical indicator of disease severity, thus allowing physicians to assess patient conditions more frequently and adjust treatment. This greatly complements conventional in-clinic medical examination that requires patients' physical presence in a specialized clinic, which is logistically inconvenient, costly, and far less frequent. The challenge, however, is the need for a patient specific predictive model to address patient heterogeneity. Training a patient-specific model suffers from small sample sizes. This can be potentially tackled by multi-task learning (MTL), which builds models for multiple related tasks (e.g., patients with the same disease) jointly to allow the models to borrow strength from each other. Existing MTL models do not suffice because they typically assume sample independence, but the samples in a telemonitoring application correspond to repeated measurements over time for the same patient that have an inherent correlation. This special data characteristic requires a linear mixed model (LMM), which has not been integrated with MTL in the existing literature. We propose a new Multi-task Linear Mixed Model (MultiLMM) model that integrates MTL and LMM in a single framework. Our methodological contributions include a mathematical formulation for MultiLMM, an efficient and converging algorithm for parameter estimation, and a theoretical analysis to reveal the reason why MultiLMM outperforms LMM by integrating the multi-task learning capability. Our simulation studies demonstrate better performance of MultiLMM than LMM under various scenarios. Finally, we present an application of using MultiLMM to predict the Unified Parkinson Disease Rating Scale (UPDRS), a clinical instrument for measuring PD severity, based on tapping signals collected by the mPower app installed on patients' smartphones. MultiLMM shows higher prediction accuracy than a collection of competing approaches under different training sample sizes.