There is a growing necessity for edge training to adapt to dynamically changing environment. Neuromorphic computing represents a significant pathway for high-efficiency intelligent computation in energy-constrained edges, but existing neuromorphic architectures lack the ability of directly training spiking neural networks (SNNs) based on backpropagation. We develop a multi-core neuromorphic architecture with Feedforward-Propagation, Back-Propagation, and Weight-Gradient engines in each core, supporting high efficient parallel computing at both the engine and core levels. It combines various data flows and sparse computation optimization by fully leveraging the sparsity in SNN training, obtaining a high energy efficiency of 1.05TFLOPS/W@ FP16 @ 28nm, 55 ~ 85% reduction of DRAM access compared to A100 GPU in SNN trainings, and a 20-core deep SNN training and a 5-worker federated learning on FPGAs. Our study develops the first multi-core neuromorphic architecture supporting the direct SNN training, facilitating the neuromorphic computing in edge-learnable applications.