Hint 1

As discussed in the lectures, the apparent gravitational field at a distance $r$ from the axis of rotation is

$${\bf g}_{\rm eff}=-g{\bf e}_3+r\omega^2\hat{\bf r}.$$

At equilibrium, this vector will be normal to the surface of the water. By considering an infinitessimal tangent vector to the surface, ${\rm d}{\bf x}=\hat{\bf r}\,{\rm d}r+{\bf e}_3\,{\rm d}x_3$, show that

$${{\rm d}x_3\over{\rm d}r}={r\omega^2\over g},$$

and hence integrate to get the parabola

$$x_3={r^2\omega^2\over 2g}+C.$$