In this paper we propose a multi-parameter reconfigurable architecture framework for patient-specific medical monitoring. This architecture is mainly composed of a set of heterogeneous processing engines and flexible communication interfaces, which enable the run-time configuration of the architecture for optimal diagnosis of different diseases. The flexibility of the proposed framework is evaluated by demonstrating two different medical applications for monitoring brain and heart status on an FPGA-based hardware prototype. The evaluated epileptic seizure detection application gains a high detection performance with overall accuracy of 98.52% and sensitivity of 99.47%. For the cardiac ICU monitoring application, the experimental results for detecting abnormality of blood pressure and heart rate in selected patients show a high true positive rate of 94.74%. By applying algorithmic enhancements in the detection scheme, we even achieve early detection of abnormalities in blood pressure in the range of few minutes before standard ICU monitor alarms with a true positive rate of 64%. With a balanced mixture of flexibility, patient-specificity, and detection accuracy at small hardware footprint, the proposed architecture can be an attractive framework for embedded monitoring of a wide variety of medical conditions.