  This is the first paper in a series of two papers. In this paper we construct
complexes of invariant differential operators which live on homogeneous spaces
of $|2|$-graded parabolic geometries of some particular type. We call them
$k$-Dirac complexes. More explicitly, we will show that each $k$-Dirac complex
arises as the direct image of a relative BGG sequence and so this fits into the
scheme of the Penrose transform. We will also prove that each $k$-Dirac complex
is formally exact, i.e., it induces a long exact sequence of infinite
(weighted) jets at any fixed point. In the second part of the series we use
this information to show that each $k$-Dirac complex is exact at the level of
formal power series at any point and that it descends to a resolution of the
$k$-Dirac operator studied in Clifford analysis.
