It is known that, if \(\phi\) is an parabolic (respectively, hyperbolic) automorphism on the unit disk \(\mathbb D\), that is to say that the angular derivative at the Denjoy-Wolff point \(\phi'(w)=1\) (respectively, \(0<\phi'(w)<1\)), then there exists \(\sigma:\mathbb D\to \{\operatorname{Re} z>0\}\) analytic such that \(\sigma\circ \phi=\sigma +i b\) for some nonzero real number \(b\) (respectively, \(\sigma\circ \phi=K\sigma\) for some \(K>1\)), see C. Pommerenke, [“On the iteration of analytic functions in a halfplane. I”, J. Lond. Math. Soc., II. Ser. 19, 439–447 (1979; Zbl 0398.30014)]. The author first observes that in the above result the map \(\sigma\) can be chosen so that its radial limit function has purely imaginary values almost everywhere. Making use of this extra fact and some other technical tools, it is shown that the numerical range of a composition operator whose symbol \(\phi\) is an inner function of parabolic type is a circular disk centered at the origin and radius larger than 1, and in particular it has point spectrum with circular symmetry. In the case of inner functions of hyperbolic type, making this time use of certain considerations on Aleksandrov-Clark measures, the author also shows that the numerical range \(W(C_\phi)\) is again a disk (open or closed) centered at the origin with radius larger than 1.