  Berget and Rhoades asked whether the permutation representation obtained by
the action of $S_{n-1}$ on parking functions of length $n-1$ can be extended to
a permutation action of $S_{n}$. We answer this question in the affirmative. We
realize our module in two different ways. The first description involves binary
Lyndon words and the second involves the action of the symmetric group on the
lattice points of the trimmed standard permutahedron.
