At CRYPTO 2019, Gohr presented a key recovery attack on SPECK32/64 assisted by deep learning. For the shortcoming that this technology cannot be used for large-state block ciphers, Chen et al. proposed a deep learning-assisted multi-stage key recovery framework in 2022, based on which key recovery attacks on large-state SPECK variants were successfully mounted. In this paper, we propose a parallelizable multi-stage key recovery framework. This framework uses the neural distinguisher trained by a new strategy to reduce the time required for the attack while maintaining the accuracy of key recovery. We conduct key recovery attacks on round-reduced SPECK64/96 and SPECK96/96. The results indicate that our framework significantly reduces the time complexity. Additionally, we train neural distinguishers over more rounds on partial bits by filtering the input differences, including more ciphertext information in training samples, using a stronger neural network and staged train method. Consequently, we obtain a set of neural distinguishers for 7-round SPECK64 and successfully extend the neural network-based key recovery attacks on SPECK64/96 by one round.