Log parsing transforms raw logs into structured templates containing constants and variables. It underpins anomaly detection, failure diagnosis, and other AIOps tasks. Current parsers are mostly reactive and log-centric. They only infer templates from logs, mostly overlooking the source code. This restricts their capacity to grasp dynamic log structures or adjust to evolving systems. Moreover, per-log LLM inference is too costly for practical deployment. In this paper, we propose LLM-SrcLog, a proactive and unified framework for log template parsing. It extracts templates directly from source code prior to deployment and supplements them with data-driven parsing for logs without available code. LLM-SrcLog integrates a cross-function static code analyzer to reconstruct meaningful logging contexts, an LLM-based white-box template extractor with post-processing to distinguish constants from variables, and a black-box template extractor that incorporates data-driven clustering for remaining unmatched logs. Experiments on two public benchmarks (Hadoop and Zookeeper) and a large-scale industrial system (Sunfire-Compute) show that, compared to two LLM-based baselines, LLM-SrcLog improves average F1-score by 2-17% and 8-35%. Meanwhile, its online parsing latency is comparable to data-driven methods and about 1,000 times faster than per-log LLM parsing. LLM-SrcLog achieves a near-ideal balance between speed and accuracy. Finally, we further validate the effectiveness of LLM-SrcLog through practical case studies in a real-world production environment.