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In this paper, we address the problem of privacy-preserving federated neural network training with $N$ users. We present Hercules, an efficient and high-precision training framework that can tolerate collusion of up to $N-1$ users. Hercules follows the POSEIDON framework proposed by Sav et al. (NDSS'21), but makes a qualitative leap in performance with the following contributions: (i) we design a novel parallel homomorphic computation method for matrix operations, which enables fast Single Instruction and Multiple Data (SIMD) operations over ciphertexts. For the multiplication of two $h\times h$ dimensional matrices, our method reduces the computation complexity from $O(h^3)$ to $O(h)$. This greatly improves the training efficiency of the neural network since the ciphertext computation is dominated by the convolution operations; (ii) we present an efficient approximation on the sign function based on the composite polynomial approximation. It is used to approximate non-polynomial functions (i.e., ReLU and max), with the optimal asymptotic complexity. Extensive experiments on various benchmark datasets (BCW, ESR, CREDIT, MNIST, SVHN, CIFAR-10 and CIFAR-100) show that compared with POSEIDON, Hercules obtains up to 4% increase in model accuracy, and up to 60$\times$ reduction in the computation and communication cost.