  A submanifold of a Riemannian symmetric space is called parallel if its
second fundamental form is a parallel section of the appropriate tensor bundle.
We classify parallel submanifolds of the Grassmannian $\rmG^+_2(\R^{n+2})$
which parameterizes the oriented 2-planes of the Euclidean space $\R^{n+2}$\,.
Our main result states that every complete parallel submanifold of
$\rmG^+_2(\R^{n+2})$\,, which is not a curve, is contained in some totally
geodesic submanifold as a symmetric submanifold. This result holds also if the
ambient space is the non-compact dual of $\rmG^+_2(\R^{n+2})$\,.
