  In this short note, we investigate the following question of Panyushev : ``Is
the sum of the index of a parabolic subalgebra of a semisimple Lie algebra
$\mathfrak{g}$ and the index of its nilpotent radical always greater than or
equal to the rank of $\mathfrak{g}$?''. Using the formula for the index of
parabolic subalgebras conjectured by Tauvel and the author, and proved by
Millet-Fauquant and Joseph, we give a positive answer to this question.
Moreover, we also obtain a necessary and sufficient condition for this sum to
be equal to the rank of $\mathfrak{g}$. This provides new examples of direct
sum decomposition of a semisimple Lie algebra verifying the ``index additivity
condition'' as stated by Ra{\"\i}s.
