Let \(G\) be a complex connected semisimple Lie group with parabolic subgroup \(P\). Let \((P,P)\) be its commutator subgroup. The generalized Borel-Weil theorem on flag manifolds has an analogous result on the Dolbeault cohomology \(H^{0,q}(G/(P,P))\). Consequently, the dimension of \(H^{0,q}(G/(P,P))\) is either 0 or \(\infty\). We show that the Dolbeault operator \(\overline\partial\) has closed image, and apply the Peter-Weyl theorem to show how \(q\) determines the value 0 or \(\infty\). For the case when \(P\) is maximal, we apply our result to compute the Dolbeault cohomology of certain examples, such as the punctured determinant bundle over the Grassmannian.