  Let $U_\zeta$ be a Lusztig quantum enveloping algebra associated to a complex
semisimple Lie algebra $\mathfrak g$ and a root of unity $\zeta$. When $L,L'$
are irreducible $U_\zeta$-modules having regular highest weights, the dimension
of $\operatorname{Ext}^n_{U_\zeta}(L,L')$ can be calculated in terms of the
coefficients of appropriate Kazhdan-Lusztig polynomials associated to the
affine Weyl group of $U_\zeta$. This paper shows for $L,L'$ irreducible modules
in a singular block that $\dim\operatorname{Ext}^n_{U_\zeta}(L,L')$ is
explicitly determined using the coefficients of parabolic Kazhdan-Lusztig
polynomials. This also computes the corresponding cohomology for $q$-Schur
algebras and many generalized $q$-Schur algebras. The result depends on a
certain parity vanishing property which we obtain from the Kazhdan-Lusztig
correspondence and a Koszul grading of Shan-Varagnolo-Vasserot for the
corresponding affine Lie algebra.
