  Let $G'$ be a connected reductive group over $\mathbb{Q}$ such that $G =
G'/\mathbb{Q}_p$ is quasi-split, and let $Q \subset G$ be a parabolic subgroup.
We introduce parahoric overconvergent cohomology groups with respect to $Q$,
and prove a classicality theorem showing that the small slope parts of these
groups coincide with those of classical cohomology. This allows the use of
overconvergent cohomology at parahoric, rather than Iwahoric, level, and
provides flexible lifting theorems that appear to be particularly well-adapted
to arithmetic applications. When $Q$ is the Borel, we recover the usual theory
of overconvergent cohomology, and our classicality theorem gives a stronger
slope bound than in the existing literature. We use our theory to construct
$Q$-parabolic eigenvarieties, which parametrise $p$-adic families of systems of
Hecke eigenvalues that are finite slope at $Q$, but that allow infinite slope
away from $Q$.
