Log parsing serves as an essential prerequisite for various log analysis tasks. Recent advancements in this field have improved parsing accuracy by leveraging the semantics in logs through fine-tuning large language models (LLMs) or learning from in-context demonstrations. However, these methods heavily depend on labeled examples to achieve optimal performance. In practice, collecting sufficient labeled data is challenging due to the large scale and continuous evolution of logs, leading to performance degradation of existing log parsers after deployment. To address this issue, we propose LUNAR, an unsupervised LLM-based method for efficient and off-the-shelf log parsing. Our key insight is that while LLMs may struggle with direct log parsing, their performance can be significantly enhanced through comparative analysis across multiple logs that differ only in their parameter parts. We refer to such groups of logs as Log Contrastive Units (LCUs). Given the vast volume of logs, obtaining LCUs is difficult. Therefore, LUNAR introduces a hybrid ranking scheme to effectively search for LCUs by jointly considering the commonality and variability among logs. Additionally, LUNAR crafts a novel parsing prompt for LLMs to identify contrastive patterns and extract meaningful log structures from LCUs. Experiments on large-scale public datasets demonstrate that LUNAR significantly outperforms state-of-the-art log parsers in terms of accuracy and efficiency, providing an effective and scalable solution for real-world deployment. \footnote{The code and data are available at \url{https://github.com/Jun-jie-Huang/LUNAR}}.