System-generated logs are typically converted into categorical log templates through parsing. These templates are crucial for generating actionable insights in various downstream tasks. However, existing parsers often fail to capture fine-grained template details, leading to suboptimal accuracy and reduced utility in downstream tasks requiring precise pattern identification. We propose a character-level log parser utilizing a novel neural architecture that aggregates character embeddings. Our approach estimates a sequence of binary-coded decimals to achieve highly granular log templates extraction. Our low-resource character-level parser, tested on revised Loghub-2k and a manually annotated industrial dataset, matches LLM-based parsers in accuracy while outperforming semantic parsers in efficiency.