  Manifolds endowed with a parabolic geometry in the sense of Cartan come with
natural sequences of differential operators and their analysis provide the so
called (curved) BGG sequence of {\v C}ap, Slov{\'a}k and Sou{\v c}ek. The
sequences involved do not form an elliptic complex in the sense of Atiyah but
enjoy similar properties. The proper framework to study these operators is the
filtered calculus associated to the natural filtration of the tangent bundle
induced by the parabolic geometry. Such analysis was carried over by Dave and
Haller in a very general setting. In this article we use their methods
associated with the transversal index theory for filtered manifolds developped
by the author in a previous paper to derive curved BGG sequences for foliated
manifolds with transverse parabolic geometry.
