Here is a partial result. Let $X$ be a smooth projective surface admitting a genus-2 fibration $\pi : X \to \mathbb{P}^1$ whose singular fibers are nodal. Then $\pi_{\ast}\omega_{X|\mathbb{P}^1}$ is a rank-2 vector bundle on $\mathbb{P}^1$, and we have a morphism $f : X \to \mathbb{P}(\pi_{\ast}\omega_{X|\mathbb{P}^1})$ of $\mathbb{P}^1-$varieties which is a branched double covering. The fact that the singular fibers of $\pi$ are all nodal implies that the branch curve $B \subset \mathbb{P}(\pi_{\ast}\omega_{X|\mathbb{P}^1})$ of $f$ is smooth. Applying Riemann-Hurwitz to each component of $B$, we see that if $\pi$ has an odd number of singular fibers, at least one of them has 2 or more nodes.

Here is a partial result. Let $X$ be a smooth projective surface admitting a genus-2 fibration $\pi : X \to \mathbb{P}^1$ whose singular fibers are all irreducible and nodal. Then $\pi_{\ast}\omega_{X|\mathbb{P}^1}$ is a rank-2 vector bundle on $\mathbb{P}^1$, and we have a morphism $f : X \to \mathbb{P}(\pi_{\ast}\omega_{X|\mathbb{P}^1})$ of $\mathbb{P}^1-$varieties which is a branched double covering. The fact that the singular fibers of $\pi$ are all nodal implies that the branch curve $B \subset \mathbb{P}(\pi_{\ast}\omega_{X|\mathbb{P}^1})$ of $f$ is smooth. Applying Riemann-Hurwitz to each component of $B$, we see that if $\pi$ has an odd number of singular fibers which are all irreducible and nodal, then at least one of them has 2 or more nodes.