Complex interactions between neurons can be simulated using multi-layer networks, such as cross-layer coupling between neurons in different brain regions. This study employs numerical calculations to investigate the synchronization characteristics of multi-layer optical pulse neural networks based on vertical-cavity surface-emitting lasers with saturable absorber (VCSEL-SA). Considering the limited signal transmission speed in neural systems, we comprehensively evaluate the impact of intra-layer and inter-layer delays on network synchronization. Our findings reveal that different coupling delays effectively induce transitions in network synchronization patterns. Furthermore, we examine the influence of key VCSEL-SA parameters on synchronization stability and demonstrate the robustness of neuronal synchronization against parameter mismatches between different layers. Finally, we validate the universality of our conclusions through a three-layer photonic neuron network. This work presents a systematic and in-depth investigation of synchronization characteristics in multi-layer networks composed of photonic neurons. The results provide valuable insights for practical applications of brain-inspired optical neural networks.