  We classify projective functors on the regular block of Rocha-Caridi's
parabolic version of the BGG category $\mathcal{O}$ in type $A$. In fact, we
show that, in type $A$, the restriction of an indecomposable projective functor
from $\mathcal{O}$ to the parabolic category is either indecomposable or zero.
As a consequence, we obtain that projective functors on the parabolic category
$\mathcal{O}$ in type $A$ are completely determined, up to isomorphism, by the
linear transformations they induce on the level of the Grothendieck group,
which was conjectured by Stroppel in \cite{St}.
