An explicit example of such a family with $1$-dimensional basis can be found in my paper  [*On surfaces of general type with $p_g=q=1, K^2=3$*][1]. 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:

> 
 2. the pencil $|G|$ is base point free;
 3. $|G|$ is the only genus $2$ pencil on $S$;
 4. $|G|$ 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 $f(E, 4)$*. 




  [1]: http://arxiv.org/abs/math/0503273