  The strong Macdonald theorems state that, for $L$ reductive and $s$ an odd
variable, the cohomology algebras $H^*(L[z]/z^N)$ and $H^*(L[z,s])$ are freely
generated, and describe the cohomological, $s$-, and $z$-degrees of the
generators. The resulting identity for the $z$-weighted Euler characteristic is
equivalent to Macdonald's constant term identity for a finite root system. We
calculate $H^*(\mathfrak{p} / z^N \mathfrak{p})$ and $H^*(\mathfrak{p}[s])$ for
$\mathfrak{p}$ a standard parahoric in a twisted loop algebra, giving strong
Macdonald theorems that take into account both a parabolic component and a
possible diagram automorphism twist. In particular we show that
$H^*(\mathfrak{p} / z^N \mathfrak{p})$ contains a parabolic subalgebra of the
coinvariant algebra of the fixed-point subgroup of the Weyl group of $L$, and
thus is no longer free. We also prove a strong Macdonald theorem for
$H^*(\mathfrak{b}; S^* \mathfrak{n}^*)$ and $H^*(\mathfrak{b} / z^N
\mathfrak{n})$ when $\mathfrak{b}$ and $\mathfrak{n}$ are Iwahori and nilpotent
subalgebras respectively of a twisted loop algebra. For each strong Macdonald
theorem proved, taking $z$-weighted Euler characteristics gives an identity
equivalent to Macdonald's constant term identity for the corresponding affine
root system. As part of the proof, we study the regular adjoint orbits for the
adjoint action of the twisted arc group associated to $L$, proving an analogue
of the Kostant slice theorem.
