We introduce a distributed quantum-classical framework that synergizes photonic quantum neural networks (QNNs) with matrix-product-state (MPS) mapping to achieve parameter-efficient training of classical neural networks. By leveraging universal linear-optical decompositions of $M$-mode interferometers and photon-counting measurement statistics, our architecture generates neural parameters through a hybrid quantum-classical workflow: photonic QNNs with $M(M+1)/2$ trainable parameters produce high-dimensional probability distributions that are mapped to classical network weights via an MPS model with bond dimension $χ$. Empirical validation on MNIST classification demonstrates that photonic QT achieves an accuracy of $95.50\% \pm 0.84\%$ using 3,292 parameters ($χ= 10$), compared to $96.89\% \pm 0.31\%$ for classical baselines with 6,690 parameters. Moreover, a ten-fold compression ratio is achieved at $χ= 4$, with a relative accuracy loss of less than $3\%$. The framework outperforms classical compression techniques (weight sharing/pruning) by 6--12\% absolute accuracy while eliminating quantum hardware requirements during inference through classical deployment of compressed parameters. Simulations incorporating realistic photonic noise demonstrate the framework's robustness to near-term hardware imperfections. Ablation studies confirm quantum necessity: replacing photonic QNNs with random inputs collapses accuracy to chance level ($10.0\% \pm 0.5\%$). Photonic quantum computing's room-temperature operation, inherent scalability through spatial-mode multiplexing, and HPC-integrated architecture establish a practical pathway for distributed quantum machine learning, combining the expressivity of photonic Hilbert spaces with the deployability of classical neural networks.