Let $\mathfrak{g}$ be a simple complex Lie algebra and let $\Delta$ be a system of positive roots relative a choice of Cartan subalgebra and $\mathfrak{b}$ the corresponding Borel subalgebra. Let $B<G$ the corresponding Borel subgroup in the connected Lie group $G$ with Lie algebra $\mathfrak{g}.$ Then, there is a theorem of Kostant which states the following:

Let $V=\{u + \sum_{\alpha \in \Delta} v_{\alpha} \ | u \in \mathfrak{b}, 0\neq v_{\alpha}\in \mathfrak{g}_{-\alpha}\}$ where $\mathfrak{g}_{-\alpha}$ is the root space corresponding to the negative simple root $-\alpha.$

Then, there exists an affine subspace $L\subset V$ such that the mapping $\phi: B\times L\rightarrow V$ defined by $\phi(b, v)=Ad(b)(v)$ is an isomorphism of algebraic varieties.

Is there a generalization of this theorem to parabolic subgroups/subalgebras other than the Borel? If there is not a generalization, is it because no such result can be true? I feel like there's probably a discussion of this somewhere in the vast literature on Lie algebras, but I haven't been able to turn anything up with google searches. Thank you very much for any help you can offer.