  Let $\mathbf{G}$ be a connected reductive group over an algebraically closed
field $\mathbb{F}$ of good characteristic, satisfying some mild conditions. In
this paper we relate tilting objects in the heart of Bezrukavnikov's exotic
t-structure on the derived category of equivariant coherent sheaves on the
Springer resolution of $\mathbf{G}$, and Iwahori-constructible
$\mathbb{F}$-parity sheaves on the affine Grassmannian of the Langlands dual
group. As applications we deduce in particular the missing piece for the proof
of the Mirkovic-Vilonen conjecture in full generality (i.e. for good
characteristic), a modular version of an equivalence of categories due to
Arkhipov-Bezrukavnikov-Ginzburg, and an extension of this equivalence.
