The number and dynamic nature of web and mobile applications presents significant challenges for assessing their compliance with data protection laws. In this context, symbolic and statistical Natural Language Processing (NLP) techniques have been employed for the automated analysis of these systems' privacy policies. However, these techniques typically require labor-intensive and potentially error-prone manually annotated datasets for training and validation. This research proposes the application of Large Language Models (LLMs) as an alternative for effectively and efficiently extracting privacy practices from privacy policies at scale. Particularly, we leverage well-known LLMs such as ChatGPT and Llama 2, and offer guidance on the optimal design of prompts, parameters, and models, incorporating advanced strategies such as few-shot learning. We further illustrate its capability to detect detailed and varied privacy practices accurately. Using several renowned datasets in the domain as a benchmark, our evaluation validates its exceptional performance, achieving an F1 score exceeding 93%. Besides, it does so with reduced costs, faster processing times, and fewer technical knowledge requirements. Consequently, we advocate for LLM-based solutions as a sound alternative to traditional NLP techniques for the automated analysis of privacy policies at scale.