An explicit example of such a family with $1$-dimensional basis can be found in my paper On surfaces of general type with $p_g=q=1, K^2=3$. The relevant statement is Proposition 6.1, that I will restate here for the reader's convenience.
Proposition. There exists exactly one irreducible family of minimal surfaces of general type with $p_g=q=1$, $K^2=3$ and a rational pencil $|G|$ of curves of genus $2$, and this family is parametrized by the coarse moduli space of elliptic curves. Moreover:
- the pencil $|G|$ is base point free;
- $|G|$ is the only genus $2$ pencil on $S$;
- $|G|$ contains $13$ singular fibres; six of these are genus $1$ curves with an ordinary double point, the other seven consist of two smooth elliptic curves intersecting transversally at a single point.
As far as I know this family was first constructed by Xiao Gang in his book Surfaces fibrées en courbes de genre deux (Lecture Notes in Mathematics 1137, Springer 1985), see in particular Chapter 3. The costruction follows from the classification of non-isotrivial genus $2$ fibrations $f \colon S \to C$ ($S$ smooth surface, $C$ smooth curve) whose associated Jacobian fibration has a fixed part $E \times C$.
In Xiao's book one can also find many other examples, see in particular the table at page $52$. The family of surfaces in the Proposition corresponds to the case $d=4$ in that table. These surfaces are explicitly mentioned in the Corollaire $4$, page $51$, where they are called fibrations $f(E, 4)$.