While reading this paper of Kollár, the following question came up. If $f:X\to Y$ is a fiber space (i.e. surjective holomorphic map with connected fibers) with $X,Y$ smooth projective manifolds, with $K_X\sim_{\mathbb{Q}} 0$, and with $f$ a submersion everywhere, does there exist a finite étale cover $Y'\to Y$ such that $X\times_{Y} Y'$ is isomorphic to a product family (and so $K_Y\sim_{\mathbb{Q}}0$ too)?

Note that since $f$ is a submersion, every fiber $F$ satisfies $K_F\sim_{\mathbb{Q}}0$.

The case when $\mathrm{dim}X=\mathrm{dim}Y+1$ (which includes the case when $\mathrm{dim}X=2$) is easy, since then the fibers are elliptic curves, and the $j$-invariant of each fiber gives a holomorphic function on $Y$, hence constant. It then follows from Fisher-Grauert that $f$ is a holomorphic fiber bundle, and you can apply for example Lemma 17 in this paper of Kollár-Larsen to conclude.

This same argument shows that in arbitrary dimension it is enough to show that all the fibers of $f$ are biholomorphic.

Thanks to work of Kawamata we know that $\kappa(Y)\leq 0$. We can also apply Theorem 4.4 in this paper of Fujino-Gongyo and get that $-K_Y$ is semiample. So if $\kappa(Y)=0$ then we obtain that $K_Y\sim_{\mathbb{Q}}0$. At this point one can apply the arguments in Theorem 4.8 of this paper of Ambro and conclude.

So the question is reduced to the following problem: is it possible to have a holomorphic submersion $f:X\to Y$ with connected fibers, where $X,Y$ are smooth projective manifolds with $K_X\sim_{\mathbb{Q}} 0$ and $\kappa(Y)<0$ ?

If we assume that $Y$ is simply connected and that $X$ has $K_X\cong\mathcal{O}_X$ and $h^{p,0}(X)=0$ for $p=1,...,\mathrm{dim} X-1$ (so $X$ is Calabi-Yau in the stronger sense), then Corollary 2.5 in this paper of Zhang-Zuo implies that this is impossible.

In general I don't know the answer to this, but when $f$ is not assumed to be a submersion (so there might be singular fibers) then $\kappa(Y)<0$ can happen, for example when $X$ is an elliptically fibered $K3$ surface and $Y=\mathbb{P}^1$.