Parameter-efficient fine-tuning often selects trainable parameters before adaptation using architectural heuristics, without accounting for their varying importance during training. We introduce \textbf{FisherAdapTune}, which progressively selects parameter groups based on temporal drift in their Fisher information. Under a local Gaussian approximation, we bound the divergence between the fine-tuned posterior and pretrained prior by accumulated Fisher-weighted update costs, motivating curvature-aware selection. FisherAdapTune measures Jensen-Shannon distance between successive Fisher-value distributions and uses an adaptive threshold to freeze stabilized groups. Across VTAB-1k classification tasks, it achieves the highest macro Top-1 accuracy among the compared methods with a smaller average trainable set than full fine-tuning. Across four segmentation backbones, it maintains competitive in-distribution performance and improves zero-shot transfer in several settings. Its selections reveal architecture-dependent patterns where input, output, and normalization parameters can remain trainable as attention and MLP groups freeze at different rates. The results support Fisher structural drift as a task-dependent signal for allocating updates during adaptation. We release our \href{https://github.com/AtlasAnalyticsLab/FisherAdapTune}{code} publicly to enable further application of our proposed approach.

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