Parkinson's disease (PD), the second most prevalent neurodegenerative disorder worldwide, frequently presents with early-stage speech impairments. Recent advancements in Artificial Intelligence (AI), particularly deep learning (DL), have significantly enhanced PD diagnosis through the analysis of speech data. Nevertheless, the progress of research is restricted by the limited availability of publicly accessible speech-based PD datasets, primarily due to privacy concerns. The goal of this systematic review is to explore the current landscape of speech-based DL approaches for PD classification, based on 33 scientific works published between January 2020 and March 2024. We discuss their available resources, capabilities, and potential limitations, and issues related to bias, explainability, and privacy. Furthermore, this review provides an overview of publicly accessible speech-based datasets and open-source material for PD. The DL approaches identified are categorized into end-to-end (E2E) learning, transfer learning (TL), and deep acoustic feature extraction (DAFE). Among E2E approaches, Convolutional Neural Networks (CNNs) are prevalent, though Transformers are increasingly popular. E2E approaches face challenges such as limited data and computational resources, especially with Transformers. TL addresses these issues by providing more robust PD diagnosis and better generalizability across languages. DAFE aims to improve the explainability and interpretability of results by examining the specific effects of deep features on both other DL approaches and more traditional machine learning (ML) methods. However, it often underperforms compared to E2E and TL approaches.