Ultra-low power operation and extreme energy efficiency are strong requirements for a number of high-growth application areas requiring near-sensor processing, including elaboration of biosignals. Parallel near-threshold computing is emerging as an approach to achieve significant improvements in energy efficiency while overcoming the performance degradation typical of low-voltage operations. In this paper, we demonstrate the capabilities of the PULP (Parallel Ultra-Low Power) platform on an algorithm for seizure detection, representative of a wide range of EEG signal processing applications. Starting from the 28-nm FD-SOI (Fully Depleted Silicon On Insulator) technology implementation of the third embodiment of the PULP architecture, we analyze the energy-efficient implementation of the seizure detection algorithm on PULP. The proposed parallel implementation exploits the dynamic voltage and frequency scaling capabilities, as well as the embedded power knobs of the PULP platform, reducing energy consumption for a seizure detection by up to 10× with respect to a sequential implementation at the nominal supply voltage and by 4.2× with respect to a sequential implementation with voltage scaling. Moreover, we analyze the trans-precision optimization of the algorithm on PULP, by means of a hybrid fixed-and floating-point implementation. This approach reduces the energy consumption by up to 43% with respect to the plain fixed-point and floating-point implementations, leveraging the requirements in terms of the precision of the kernels composing the processing chain to improve energy efficiency. Thanks to the proposed architecture and system-level approach for optimization, we demonstrate that PULP reduces energy consumption by up to 140× with respect to commercial low-power microcontrollers, being able to satisfy the real-time constraints typical of bio-medical applications, breaking the barrier of microwatts for a 50-ms complete seizure detection and a few milliwatts for a 5-ms detection latency on a fully-programmable architecture.