Modern software systems generate massive volumes of runtime logs, necessitating efficient and accurate log parsing to enable critical downstream tasks such as anomaly detection and root cause analysis. Recently, large language models (LLMs) have achieved advanced accuracy on log parsing, but their deployment in production environments faces two major limitations: (1) the privacy risks associated with commercial LLMs, driving the adoption of local deployment, and (2) the stringent latency and throughput requirements imposed by high-volume log streams, which existing LLM-based parsers fail to meet. Although recent efforts have reduced the number of LLM queries, they overlook the high latency of the LLM invocations, where concurrent log parsing requests can cause serve performance degradation of LLM inference system. In this study, we present InferLog, the first LLM inference optimization method for online log parsing. Our key insight is that the inference efficiency emerges as the vital bottleneck in LLM-based online log parsing, rather than parsing accuracy. InferLog accelerates inference by designing (1) A Prefix-aware ICL Refinement policy to refine the examples and permutation of in-context learning to improve the prefix caching efficiency. (2) A rapid and task-specific configuration tuning pipeline based on meta-learning to find the optimal LLM scheduling-related configuration for dynamic log parsing workloads. The experimental results based on Loghub dataset and vLLM demonstrate that InferLog significantly outperforms existing inference optimization methods and markedly accelerates the state-of-the-art LLM-based log parser without compromising parsing accuracy.