In this paper, we propose a quasi-periodic parallel WaveGAN (QPPWG) waveform generative model, which applies a quasi-periodic (QP) structure to a parallel WaveGAN (PWG) model using pitch-dependent dilated convolution networks (PDCNNs). PWG is a small-footprint GAN-based raw waveform generative model, whose generation time is much faster than real time because of its compact model and non-autoregressive (non-AR) and non-causal mechanisms. Although PWG achieves high-fidelity speech generation, the generic and simple network architecture lacks pitch controllability for an unseen auxiliary fundamental frequency ($F_{0}$) feature such as a scaled $F_{0}$. To improve the pitch controllability and speech modeling capability, we apply a QP structure with PDCNNs to PWG, which introduces pitch information to the network by dynamically changing the network architecture corresponding to the auxiliary $F_{0}$ feature. Both objective and subjective experimental results show that QPPWG outperforms PWG when the auxiliary $F_{0}$ feature is scaled. Moreover, analyses of the intermediate outputs of QPPWG also show better tractability and interpretability of QPPWG, which respectively models spectral and excitation-like signals using the cascaded fixed and adaptive blocks of the QP structure.