**Background**

Let $F$ be a $p$-adic local field, and let $G$ be a connected reductive group over $F$. Recall that there is a rich theory of compact open subgroups of $G(F)$ which is, essentially, encapsulated in the theory of Bruhat–Tits. There they associate to $G$ a combinatorial object called a building $B(G,F)$ upon which $G(F)$-actions, and for which many classes of 'nice' compact open subgroups of $G(F)$ can be understood as stabilizers of certain 'nice' combinatorial subobjects of $B(G,F)$. These are incredibly valuable since these 'nice' compact open subgroups of $G(F)$ are pivotal in understanding many aspects of the structure of $G(F)$ (e.g. in the construction of representations of $G(F)$ in the form of the Moy–Prasad filtration or as 'reasonable levels' in Shimura varieties).

This combinatorial data can be somewhat daunting, and so it is sometimes (at least for me) more convenient to have a way of understanding these 'nice' compact open subgroups of $G(F)$ without needing to understand the inner workings of the Bruhat–Tits machine.

As an example, one type of 'nice' compact open subgroup of $G(F)$ are the so-called *hyperspecial* subgroups. One can define these purely in terms of $B(G,F)$, but one can also define them as $\mathcal{G}(\mathcal{O}_F)$ where $\mathcal{G}$ is a reductive model of $G$ over $\mathcal{O}_F$. This allows one to work with hyperspecial subgroups without having to understand the whole definition of $G(\mathcal{O}_F)$.

The question below is whether another class of 'nice' subgroups can be understand in a similar way.

**Actual question**

Let $F$ be a $p$-adic local field with ring of integers $\mathcal{O}_F$ and residue field $k$. Let $G$ be a connected unramified reductive group over $F$. To what extent is it true that parahoric subgroups of $G(F)$ are pullbacks of parabolic subgroups of $\mathcal{G}_k$ for $\mathcal{G}$ a reductive model of $G$ over $\mathcal{O}_F$?

Namely, if $\mathcal{G}$ is a reductive model of $G$ over $\mathcal{O}_F$, and $P$ is a parabolic subgroup of $\mathcal{G}_k$, then is $\mathrm{red}^{-1}(P(k))\subseteq \mathcal{G}(\mathcal{O}_F)$ a parahoric subgroup of $G(F)$ (here $\mathrm{red}:\mathcal{G}(\mathcal{O})\to \mathcal{G}(k)$ is the reduction map) a parahoric subgroup of $G(F)$? Do they all arise in this manner?

I know that one can define parahorics as the $\mathcal{O}_F$-points of 'parahoric group schemes', but the definition of these is very opaque to me. Do these generalize the above construction in some sense?

If one can say more, feel free to assume that $G$ is split.