Fundamental group of family of rational varieties

Question

Let $X$ be a smooth, projective, simply connected variety over a field $k$ (i.e. $\pi_1^{\text{et}} = 1$). Let $f: Y \to X$ be a family of rational varieties parametrized by $X$, such that $Y$ is smooth and projective. Then $Y$ should be simply connected. Is this right, and if so, how do I prove this?