Let $x \in \mathfrak{g}$ (or $x \in G$) be a nilpotent (resp. unipotent) element of a simple Lie algebra (resp. linear algebraic group). One can associate to this data a *Jacobson-Morozov parabolic subalgebra*, defined in the Lie algebra as follows. Choose a $\mathfrak{sl}_2$-triple $(x, y,h)$ and define
$$\mathfrak{p} = \bigoplus_{i \geq 0} \mathfrak{g}_i$$
where $\mathfrak{g}_i$ is the $i$th eigenspace of $h$. This is independent of the choice of $h$ (and $y$). One can then take a corresponding parabolic subgroup $P$ in the group $G$.

My question is: are there other characterizations of $P$ that don't require us to choose a triple first?

Also, there may be other parabolics of the same type as $P$ (i.e. conjugate to $P$) which contain $x$. In general, I know Springer fibers are hard to describe, but I don't know if something manageable can be said about these particular ones (and for partial flag varieties instead). And, is there something that characterizes the Jacobson-Morozov parabolic amongst these?

I'd be happy to take just $G = SL_n$ so that $G/P$ are different partial flag varieties. For example, for $G = SL_3$, both the regular and subregular orbits have Jacobson-Morozov parabolic conjugate to a Borel $B$. The regular Springer fiber is just a point (so there's only one choice). The subregular nilpotents have $\mathbb{P}^1 \cup_{\text{pt}} \mathbb{P}^1$ as a Springer fiber and the Jacobson-Morozov parabolic is the point in the intersection of the two $\mathbb{P}^1$s.