  We consider parallel submanifolds $M$ of a Riemannian symmetric space $N$ and
study the question whether $M$ is extrinsically homogeneous in $N$\,, i.e.\
whether there exists a subgroup of the isometry group of $N$ which acts
transitively on $M$\,. First, given a "2-jet" $(W,b)$ at some point $p\in N$
(i.e. $W\subset T_pN$ is a linear space and $b:W\times W\to W^\bot$ is a
symmetric bilinear form)\,, we derive necessary and sufficient conditions for
the existence of a parallel submanifold with extrinsically homogeneous tangent
holonomy bundle which passes through $p$ and whose 2-jet at $p$ is given by
$(W,b)$\,. Second, we focus our attention on complete, (intrinsically) {\em
irreducible} parallel submanifolds of $N$\,. Provided that $N$ is of compact or
non-compact type, we establish the extrinsic homogeneity of every complete,
irreducible parallel submanifold of $N$ whose dimension is at least 3 and which
is not contained in any flat of $N$\,.
