Broadband sound absorption has long been a concern in noise control engineering, but the inverse design of multi-parallel microperforated panels (MPPs) for broadband sound absorption remains challenging. To address this issue, we propose a deep learning model that combines a variational autoencoder (VAE) with a physics-informed neural network (PINN) to accelerate the inverse design process of a multi-parallel MPP. Following Maa’s theory, we generated a dataset of 500,000 samples to train the model. By incorporating the PINN, we added an acoustic physical constraint to the loss function, promoting model convergence and the derivation of stable, unified parameters. The efficacy of the inverse design model was validated through theoretical analysis, finite element simulations, and impedance tube experiments. The experimental results show that the average sound absorption coefficient of multi-parallel MPPs within the frequency range of 500–1200 Hz is 0.85. Our work contributes to accelerating the inverse design of multi-parallel acoustic metamaterials.