We characterise n th-order ODEs for which the space of solutions M is equipped with a particular paraconformal structure in the sense of Bailey and Eastwood [T.N. Bailey, M.G. Eastwood, Complex Paraconformal manifolds, their differential geometry and twistor theory, Forum Math. 3 (1991) 61–103], 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 G₃ holonomy of Bryant. If n=4, or n≥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 3M 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.