We have an infinite cylinder of radius $a$ around the $z$ axis.
The current density inside the cylinder is uniform: $\vec J=J_0\hat z$.
We wish to find the magnetic vector potential $\vec A$ (though I am not asking for the whole solution, just the specific step described below).
In cylindrical coordinates, $\vec B \parallel \hat \theta$, and therefore $(\vec\nabla\times\vec A)\parallel \hat \theta$.
My course's notes say that from this and from the symmetry of the problem, we can deduce that $\vec A\parallel \hat z$.
How exactly can we deduce that?