We construct a co-
t
t
-structure on the derived category of coherent sheaves on the nilpotent cone
N
\mathcal {N}
of a reductive group, as well as on the derived category of coherent sheaves on any parabolic Springer resolution. These structures are employed to show that the push-forwards of the “exotic parity objects” (considered by Achar, Hardesty, and Riche [Transform. Groups 24 (2019), pp. 597–657]), along the (classical) Springer resolution, give indecomposable objects inside the coheart of the co-
t
t
-structure on
N
\mathcal {N}
. We also demonstrate how the various parabolic co-
t
t
-structures can be related by introducing an analogue to the usual translation functors. As an application, we give a proof of a scheme-theoretic formulation of the relative Humphreys conjecture on support varieties of tilting modules in type
A
A
for
p
>
h
p>h
.