The relevant paper is "Local rigidity of quasi-regular varieties" by Pasquier and Perrin, Math. Z. 265 (2010), no. 3, 589–600. They construct a smooth fibration over $\mathbb{C}$ such that the fiber over $t\neq 0$ is a orthogonal grassmannian $\mathbb{G}_q(2,7)$, but the fiber over 0 is not homogeneous.

The relevant paper is "Local rigidity of quasi-regular varieties" by Pasquier and Perrin, Math. Z. 265 (2010), no. 3, 589–600. They construct a smooth fibration over $\mathbb{C}$ such that the fiber over $t\neq 0$ is a orthogonal grassmannian $\mathbb{G}_q(2,7)$, but the fiber over 0 is not homogeneous.

Edit: Actually I overestimated the question. As it stands, just take the family of quadrics given by $X^2+Y^2+Z^2+tT^2=0$ in $\mathbb{P}^3\times \mathbb{C}$. The family cannot be trivial over the punctured disk because the monodromy exchanges the two generators of $H^2$ of a smooth fiber