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During the past two decades, epileptic seizure detection and prediction algorithms have evolved rapidly. However, despite significant performance improvements, their hardware implementation using conventional technologies, such as Complementary Metal-Oxide-Semiconductor (CMOS), in power and area-constrained settings remains a challenging task; especially when many recording channels are used. In this paper, we propose a novel low-latency parallel Convolutional Neural Network (CNN) architecture that has between 2-2,800x fewer network parameters compared to SOTA CNN architectures and achieves 5-fold cross validation accuracy of 99.84% for epileptic seizure detection, and 99.01% and 97.54% for epileptic seizure prediction, when evaluated using the University of Bonn Electroencephalogram (EEG), CHB-MIT and SWEC-ETHZ seizure datasets, respectively. We subsequently implement our network onto analog crossbar arrays comprising Resistive Random-Access Memory (RRAM) devices, and provide a comprehensive benchmark by simulating, laying out, and determining hardware requirements of the CNN component of our system. To the best of our knowledge, we are the first to parallelize the execution of convolution layer kernels on separate analog crossbars to enable 2 orders of magnitude reduction in latency compared to SOTA hybrid Memristive-CMOS DL accelerators. Furthermore, we investigate the effects of non-idealities on our system and investigate Quantization Aware Training (QAT) to mitigate the performance degradation due to low ADC/DAC resolution. Finally, we propose a stuck weight offsetting methodology to mitigate performance degradation due to stuck RON/ROFF memristor weights, recovering up to 32% accuracy, without requiring retraining. The CNN component of our platform is estimated to consume approximately 2.791W of power while occupying an area of 31.255mm$^2$ in a 22nm FDSOI CMOS process.