  Let $G$ be the group of rational points of a split connected reductive group
over a non-archimedean local field of residue characteristic $p$, and let
$\mathcal{H}$ denote the pro-$p$ Iwahori-Hecke algebra of $G$ over a field of
characteristic $p$. We study the parabolic induction functor for
$\mathcal{H}$-modules in terms of the Gorenstein projective model structures
introduced by Hovey. Let $\text{Ho}(\mathcal{H})$ denote the associated
homotopy category of this model structure. We show that
$\text{Ho}(\mathcal{H})$ and its thick subcategory generated by the essential
images of finitely many parabolic induction functors are related via a
recollement of triangulated categories. We then investigate the isomorphism
classes of simple $\mathcal{H}$-modules in $\text{Ho}(\mathcal{H})$ and give a
complete classification for simple supersingulars when $G=\mathrm{GL}_n$.
