Large Language Models and Vision-Language Models have achieved impressive performance across a wide range of tasks, yet they remain vulnerable to carefully crafted perturbations. In this study, we seek to pinpoint the sources of this fragility by identifying parameters and input dimensions (pixels or token embeddings) that are susceptible to such perturbations. To this end, we propose a stability measure called FI, First order local Influence, which is rooted in information geometry and quantifies the sensitivity of individual parameter and input dimensions. Our extensive analysis across LLMs and VLMs (from 1.5B to 13B parameters) reveals that: (I) A small subset of parameters or input dimensions with high FI values disproportionately contribute to model brittleness. (II) Mitigating the influence of these vulnerable parameters during model merging leads to improved performance.