Let $G$ be a connected simple group over a local field $k$. Let $I\subset G(\mathcal{O})$ denote an Iwahori subgroup of $G(k)$ with Lie algebra $\mathfrak{i}$. Let $P\supseteq I$ be any other parahoric subgroup with Lie algebra $\mathfrak{p}$. Define the relative dimension of $\mathfrak{p}$ and $\mathfrak{g}(\mathcal{O})$ by $$ (\mathfrak{g}(\mathcal{O}): \mathfrak{p}) = \dim(\mathfrak{g}(\mathcal{O})/\mathfrak{i}) - \dim(\mathfrak{p} / \mathfrak{i}). $$
Definition: Let us call $P$ miraculous if $(\mathfrak{g}(\mathcal{O}): \mathfrak{p})=\mathrm{rank}(G)$.
For instance, when $G=\mathrm{SL}_n$, and $P$ is the parahoric obtained from the miraculous parabolic, the $P$ is indeed a miraculous parahoric.
Question: Which groups $G$ contain a miraculous parahoric? What is the corresponding Levi subgroup $L:=P/P(1)$?
My impression is that for $G=Sp_{2n}$ such a parahoric exists. For instance, for $Sp_4$, see p4 of this paper.
Note: One can also consider the case when G is quasi-split. Then G(O) should probably be interpreted as the hyperspecial parahoric.