  We study tilting and projective-injective modules in a parabolic BGG category
$\mathcal O$ for an arbitrary classical Lie superalgebra. We establish a
version of Ringel duality for this type of Lie superalgebras which allows to
express the characters of tilting modules in terms of those of simple modules
in that category. We also obtain a classification of projective-injective
modules in the full BGG category $\mathcal O$ for all simple classical Lie
superalgebras. We then classify and give an explicit combinatorial description
of parabolic subalgebras of the periplectic Lie superalgebras and apply our
results to study their tilting modules in more detail.
