In a course on Manifolds its often emphasized that [integral curves of] vector fields are the right way to do ODE theory without coordinates. I'd say that ODEs and vector fields are morally the same. My understanding is that integral sub-manifolds of distributions are essentially first-order PDEs.
But we also like to study ODEs and PDEs of arbitrary order. Do these have their own place on a manifold? There's a couple objects one could write down whose coordinate representations will involve higher derivatives, such as $\nabla_v\dot{x}$, or $(\nabla_v)^n\dot{x}$. Probably if you write down the coordinate equation for an integral curves of a section in $TTTTTT\dots TM$ it will also involve higher derivatives. Nothing stands out as nearly so obvious as in the first-order case.