In this paper, we propose Ahead-of-Time (AoT) P-Tuning, a novel parameter-efficient fine-tuning method for pre-trained Language Models (LMs) that adds input-dependent bias before each Transformer layer. We evaluate AoT P-Tuning on GLUE and SuperGLUE benchmarking datasets using RoBERTa and DeBERTa models, showing that it outperforms BitFit and is comparable or better than other baseline methods for efficient fine-tuning. Additionally, we assess the inference overhead of AoT P-Tuning and demonstrate that it introduces negligible overhead compared to established baseline methods. Our method enables multi-task inference with a single backbone LM, making it a practical solution for real-world applications.