In general, for magnetic fields it is faster to look for planes of symmetry. It usually gives more information than parity. These are planes that leave the current distribution invariant by reflection. Since the magnetic field is a pseudo vector, it is a plane of antisymmetry of the field. Thus it will necessarily be normal to this plane and you thus directly know its direction.

In your case, any plane parallel to the $xy$ plane is a plane of symmetry, so you know that the magnetic field is necessarily along $z$.

Now that you know the only non zero component, you can use the usual symmetry arguments to see that it has only a radial dependence (invariant by translation along $z$ and rotation about the $z$ axis).

Hope this helps.

In general, for magnetic fields it is faster to look for planes of symmetry. It usually gives more information than parity. These are planes that leave the current distribution invariant by reflection, assuming that the current density is a vector. Since the magnetic field is a pseudo vector, it is a plane of antisymmetry of the field. This specifies its direction since $B$ is normal to this plane.

In your case, any plane parallel to the $xy$ plane is a plane of symmetry, so the magnetic field is necessarily along $z$.

Now that you know the only non zero component, you can use the usual symmetry arguments to see that it has only a radial dependence (invariant by translation along $z$ and rotation about the $z$ axis).

Hope this helps.