  We give a full classification of higher order parallel surfaces in
three-dimensional homogeneous spaces with four-dimensional isometry group, i.e.
in the so-called Bianchi-Cartan-Vranceanu family. This gives a positive answer
to a conjecture formulated in 2002. As a partial result, we prove that totally
umbilical surfaces only exist if the space is a Riemannian product of a surface
of constant Gaussian curvature and the real line, and we give a local
parametrization of all totally umbilical surfaces.
