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.



Characterizations of Jacobson-Morozov parabolics associated to a nilpotent
It's difficult to sort out the various questions you are combining here, so it would be helpful to tighten the formulation. Taken at face value, the answer to your basic question "are there other characterizations of $P$ that don't require us to choose a triple first?" seems to be negative, though I'm not sure exactly what you are looking for. Specific examples might be helpful.


In any case your basic set-up requires characteristic 0 (or large enough characteristic). Typically not all parabolics need arise this way and, moreover, distinct orbits may have the same canonical parabolic: this is seen for example in type $G\_2$, where there are 5 nilpotent orbits but only 4 parabolics (up to conjugacy) and where 3 of the orbits lead to the same minimal parabolic but none leads to the other minimal parabolic. 


Aside from these complications, the Bala-Carter classification method for the classes or orbits (which has some advantages over the older Dynkin method) doesn't emphasize canonical parabolics. Rather, the *distinguished* unipotents/nilpotents and the corresponding *distinguished* parabolics having these as Richardson classes/orbits are essential for their inductive approach. 


I'm not sure how easy it is in practice to deal with the canonical parabolics. These differ in subtle ways from the somewhat reverse Richardson method, for example when the group centralizers involved fail to be connected (which doesn't occur in type $A\_n$ but often does occur elsewhere). 


