  We consider the inverse mean curvature flow by parallel hypersurfaces in
space forms. We show that such a flow exists if and only if the initial
hypersurface is isoparametric. The flow is characterized by an algebraic
equation satisfied by the distance function of the parallel hypersurfaces. The
solutions to the flow are obtained explicitly when the distinct principal
curvatures have the same multiplicity. This is an additional assumption only
for isoparametric hypersurfaces of the hyperbolic space or of the sphere with
two or four distinct principal curvatures. The boundaries of the maximal
interval of definition, when finite, are determined in terms of the number $g$
of distinct principal curvatures, their multiplicities $m$ and the mean
curvature $H$ of the initial hypersurface. We describe the collapsing
submanifolds of the flow at the boundaries of the interval. In particular, we
show in the Euclidean space the solutions are eternal, while in the hyperbolic
space there are eternal and immortal solutions. Starting with a connected
isoparametric submanifold of the sphere, we show that the flow is an ancient
solution, that collapses into a minimal hypersurface whose square length of its
second fundamental form and its scalar curvature are constants given in terms
of $g$ and $n$. The minimal hypersurface is totally geodesic when $g=1$, it is
a Clifford minimal hypersurface of the sphere when $g=2$ and it is a Cartan
type minimal submanifold when $g\in\{3,4,6\}$.
