Log parsing is defined as the process of extracting structured information from unstructured log data. It is an important step prior to many log analytics tasks. The emergence of Large Language Models (LLMs), like Generative Pre-trained Transformers (GPTs), has driven the development of novel log parsing methods. Existing studies have examined the effectiveness of large-scale general-purpose LLMs in log parsing. In this paper, we argue that the long-term adoption of such LLMs pose challenges of data privacy, cost, and tool integration. To address these challenges, we explore the viability of supervised fine-tuning of an open-source compact LLM for log parsing as a prospective alternative. To this end, we fine-tune the Mistral-7B-Instruct LLM on a diverse set of log files and evaluate its performance, in terms of both accuracy and robustness, against OpenAI's GPT-4-Turbo using different configuration settings. We apply two evaluation approaches, namely metric-based and LLM-based. Our overall findings show that fine-tuning a compact LLM such as Mistral-7B provides similar and sometimes better results than using a large-scale LLM, in our case GPT-4-Turbo. These findings are important because they enable companies to use a smaller LLM that they can readily adapt to parsing their log data, and integrate into their log analytics tools, without the need to rely on third-party LLM providers.