I might be misunderstanding the question, but the Dolbeault operators are usually defined in terms of the de Rham operator and the complex structure (see e.g. Wells' book or Griffith and Harris). The example you outline generalizes to the situation $G_{\mathbb{C}} / P = G / G_0$, where $G$ is compact, $G_{\mathbb{C}}$ is the complexification, $P$ is a parabolic subgroup, and $G_0 = G \cap P$. The holomorphic tangent bundle is the homogeneous vector bundle $G_\{\mathbb{C}} \times_P \mathfrak{g}\_{\mathbb{C}} / \mathfrak{p}$, and the cotangent bundle is a homogeneous vector bundle in similar fashion.

In this case, the Dolbeault complex with coefficients in a homogeneous vector bundle $G_{\mathbb{C}} \times_P V$ translates to the Koszul complex for the relative
Lie algebra cohomology $H^*(\mathfrak{g},\mathfrak{g}\_0,V \otimes C^{\infty}(G))$. The Dolbeault operator $\overline{\partial}$ translates to the boundary operator for Lie algebra cohomology, which of course involves the action of $\mathfrak{g}$. I haven't worked out what
happens to $\partial$ in this situation, but most likely a similar expression in terms of the Lie algebra can be derived. The translation works by thinking about the smooth sections as elements of $C^{\infty}(G) \otimes V)^{\mathfrak{g}\_0}$ (for holomorphic sections use $C^{\infty}(G) \otimes V)^{\mathfrak{p}}$).