Fine-tuning on task-specific question-answer pairs is a predominant method for enhancing the performance of instruction-tuned large language models (LLMs) on downstream tasks. However, in certain specialized domains, such as healthcare or harmless content generation, it is nearly impossible to obtain a large volume of high-quality data that matches the downstream distribution. To improve the performance of LLMs in data-scarce domains with domain-mismatched data, we re-evaluated the Transformer architecture and discovered that not all parameter updates during fine-tuning contribute positively to downstream performance. Our analysis reveals that within the self-attention and feed-forward networks, only the fine-tuned attention parameters are particularly beneficial when the training set's distribution does not fully align with the test set. We empirically validate TAIA using two general instruction-tuning datasets and evaluate it on seven downstream tasks involving math, reasoning, and knowledge understanding across LLMs of different parameter sizes and fine-tuning techniques.