We propose both serial and parallel proximal (linearized) alternating direction method of multipliers (ADMM) algorithms for training residual neural networks. In contrast to backpropagation-based approaches, our methods inherently mitigate the exploding gradient issue and are well-suited for parallel and distributed training through regional updates. Theoretically, we prove that the proposed algorithms converge at an R-linear (sublinear) rate for both the iteration points and the objective function values. These results hold without imposing stringent constraints on network width, depth, or training data size. Furthermore, we theoretically analyze our parallel/distributed ADMM algorithms, highlighting their reduced time complexity and lower per-node memory consumption. To facilitate practical deployment, we develop a control protocol for parallel ADMM implementation using Python's multiprocessing and interprocess communication. Experimental results validate the proposed ADMM algorithms, demonstrating rapid and stable convergence, improved performance, and high computational efficiency. Finally, we highlight the improved scalability and efficiency achieved by our parallel ADMM training strategy.