  In this paper, we study the parabolic BGG categories for graded Lie
superalgebras of Cartan type over complex numbers. The gradation of such a Lie
superalgebra $\ggg$ naturally arises, with the zero component $\ggg_0$ being a
reductive Lie algebra. We first show that there are only two proper parabolic
subalgebras containing Levi subalgebra $\ggg_0$: the ``maximal one" $\sfp_\max$
and the ``minimal one" $\sfp_\min$. Furthermore, the parabolic BGG category
arising from $\sfp_\max$, essentially turns out to be a subcategory of the one
arising from $\sfp_\min$. Such a priority of $\sfp_\min$ in the sense of
representation theory reduces the question to the study of the ``minimal
parabolic" BGG category $\comi$ associated with $\sfp_\min$. We prove the
existence of projective covers of simple objects in these categories, which
enables us to establish a satisfactory block theory. Most notably, our main
results are as follows:
  (1) We classify and obtain a precise description of the blocks of $\comi$.
  (2) We investigate indecomposable tilting and indecomposable projective
modules in $\comi$, and compute their character formulas.
