This follows from Theorem 2.9 in their paper "Schubert Cells and cohomology of the spaces G/P"

Thank you so much! This is precisely what I was looking for.

This is helpful from the nomenclature aspect, but this still only uses one of the $\mathbb{P}^1$ bundles, with no mention of the other. In particular in "$T$-equivariant $K$-theory of generalized flag varieties" by Kostant and Kumar, the $G/B \rightarrow G/P_i$ fibration is the only one I can find.

Well, since $A^.$ is a chain complex, I don't really understand what $\underline{Ext}^{i+k+1}(A^., O_X)$ is. I'm sorry, I realize in my original question that I didn't write $A^.$ consistently.

Thank you! I'm going to try to work this out in a moment. A quick follow up question: In the 2nd proposition, $A^.$ is a bounded complex of coherent sheaves; where does the chain complex degree appear in your answer?