We introduce a novel framework for privacy-preserving multi-party neural network training over $\mathbb{Z}_{2^k}$ with semi-honest security in the honest-majority setting. Our work utilizes Shamir secret sharing scheme over Galois rings $GR(2^k, d)$ and is scalable in the number of participants. Our primary contribution is a generalization of existing data packing techniques used in private training through Reverse Multiplication-Friendly Embedding (RMFE), which enables a higher packing density and thus more efficient SIMD-style parallel computation. Notably, our work is the first to support a general form of RMFE, lifting a common restriction from previous approaches. To holistically optimize the training process, we further integrate mixed-circuit techniques to be fully compatible with our RMFE-based packing scheme. This enables our protocol to efficiently compute nonlinear functions, such as comparison, by leveraging bit-wise computations over $GR(2, d)$. We consolidate these advances into an end-to-end parallel training framework. Experimental results on both fully connected and convolutional neural networks validate the practical performance advantages of our framework compared to existing methods.