Let $G$ be a reductive group scheme over some base $X$ and $P \subseteq G$ a parabolic subgroup. To a $P$-torsor $\mathscr{E}_P$, we may associate a $G$-torsor $\mathscr{E} = G \times^P \mathscr{E}_P$, which is $G \times \mathscr{E}_P$ mod the relation $(gp, s) \sim (g, ps)$, with $G$ acting by $g \cdot (h, s) = (gh, s)$. This gives a $P$-equivariant monomorphism $\mathscr{E}_P \hookrightarrow \mathscr{E}$ sending $s$ to $(1, s)$, and this gives us an inclusion of group schemes $\mathrm{Aut}(\mathscr{E}_P) \xrightarrow{\sim} \mathrm{Stab}(\mathscr{E}_P) \subseteq \mathrm{Aut}(\mathscr{E})$.

$\mathrm{Aut}(\mathscr{E}_P)$ is an inner form of $P$ (indeed, it is given by twisting $P$ by $\mathscr{E}_P$), and likewise $\mathrm{Aut}(\mathscr{E})$ is an inner form of $G$. Upon passing to an étale cover $\widetilde{X} \rightarrow X$ and choosing a trivialization $\mathscr{E}_P$ (which automatically gives a trivialization of $\mathscr{E}$ compatible with the map $\mathscr{E}_P \rightarrow \mathscr{E}$), we obtain an isomorphism $G|_{\widetilde{X}} \rightarrow \mathrm{Aut}(E)|_{\widetilde{X}}$ sending $P$ to $\mathrm{Aut}(\mathscr{E}_P)|_{\widetilde{X}}$.

This discussion defines a functor from the groupoid of $P$-torsors on $X$ to the groupoid of pairs $(\mathscr{E}, \mathscr{P})$ where $\mathscr{E}$ is a $G$-torsor on $X$ and $\mathscr{P}$ is a parabolic subgroup of $\mathrm{Aut}(\mathscr{E})$. A morphism $(\mathscr{E}, \mathscr{P}) \rightarrow (\mathscr{E}', \mathscr{P}')$ is an isomorphism of $G$-torsors $\varphi \colon \mathscr{E} \rightarrow \mathscr{E}'$ carrying $\mathscr{P}$ to $\mathscr{P}'$ under the induced map on automorphism groups (given by conjugation by $\varphi$).

Example 10.6.2 of the book *Weil's conjecture for function fields* by Gaitsgory-Lurie, it is claimed that this functor is an equivalence of categories. What's the quasi-inverse functor? I can't seem to find a proof anywhere.

If it helps, feel free to assume that $X$ is a curve over a finite field $k$ and that $G = G_0 \times_{\mathrm{Spec}(k)} X$ for an adjoint split reductive group $G_0$ over $\mathrm{Spec}(k)$.