  Every compact 3-Sasakian 7-manifold $M$ admits a canonical 2-parameter family
of co-closed $\text{G}_2$-structures $\varphi_{a,b}$ for $a,b > 0$, as well as
a foliation by $\varphi_{a,b}$-associative 3-folds whose leaf space $X$ is a
positive quaternion-K\"{a}hler 4-orbifold. We prove that associative 3-folds in
$(M,\varphi_{a,b})$ that are ruled by a certain type of geodesic are in
correspondence with pseudo-holomorphic curves in the almost-complex 8-manifold
$Z \times S^2$, where $Z$ is the twistor space of $X$ equipped with its strict
nearly-K\"{a}hler structure. As an application, we construct infinitely many
topological types of non-trivial, compact associative 3-folds in the squashed
7-spheres $(S^7, \varphi_{a,b})$ and squashed exceptional Aloff-Wallach spaces
$(N_{1,1}, \varphi_{a,b})$. Topologically, our examples are circle bundles over
a genus $g$ surface, for any $g \geq 0$.
