Compared to traditional electrodynamic loudspeakers, the parametric array loudspeaker (PAL) offers exceptional directivity for audio applications but suffers from significant nonlinear distortions due to its inherent intricate demodulation process. The Volterra filter-based approaches have been widely used to reduce these distortions, but the effectiveness is limited by its inverse filter's capability. Specifically, its pth-order inverse filter can only compensate for nonlinearities up to the pth order, while the higher-order nonlinearities it introduces continue to generate lower-order harmonics. In contrast, this paper introduces the modern deep learning methods for the first time to address nonlinear identification and compensation for PAL systems. Specifically, a feedforward variant of the WaveNet neural network, recognized for its success in audio nonlinear system modeling, is utilized to identify and compensate for distortions in a double sideband amplitude modulation-based PAL system. Experimental measurements from 250 Hz to 8 kHz demonstrate that our proposed approach significantly reduces both total harmonic distortion and intermodulation distortion of audio sound generated by PALs, achieving average reductions to 4.55% and 2.47%, respectively. This performance is notably superior to results obtained using the current state-of-the-art Volterra filter-based methods. Our work opens new possibilities for improving the sound reproduction performance of PALs.