We characterise $n$th order ODEs for which the space of solutions $M$ is
equipped with a particular paraconformal structure in the sense of \cite{BE},
that is a splitting of the tangent bundle as a symmetric tensor product of
rank-two vector bundles. This leads to the vanishing of $(n-2)$ quantities
constructed from of the ODE.
If $n=4$ the paraconformal structure is shown to be equivalent to the exotic
${\cal G}_3$ holonomy of Bryant. If $n=4$, or $n\geq 6$ and $M$ admits a
torsion--free connection compatible with the paraconformal structure then the
ODE is trivialisable by point or contact transformations respectively.
If $n=2$ or 3 $M$ admits an affine paraconformal connection with no torsion.
In these cases additional constraints can be imposed on the ODE so that $M$
admits a projective structure if $n=2$, or an Einstein--Weyl structure if
$n=3$. The third order ODE can in this case be reconstructed from the
Einstein--Weyl data.