The score in the $\psi$-parametrisation is obtained by the chain rule: \begin{align*} \frac{d}{d\psi}\log f(X, \theta(\psi)) = \frac{d}{d\theta}\log f(X, \theta)\Big|_{\theta = \theta(\psi)} \cdot \frac{d\theta}{d\psi}. \end{align*} Squaring and taking expectations under $\mathbb{P}_\theta$ (with $\theta = \theta(\psi)$): \begin{align*} I_\psi(\psi) = \mathbb{E}_\theta\!\left[\left(\frac{d}{d\theta}\log f(X,\theta)\right)^2\right] \cdot \left(\frac{d\theta}{d\psi}\right)^2 = I_\theta(\theta(\psi))\left(\frac{d\theta}{d\psi}\right)^2. \end{align*}