Parabolic subgroups of $\mathrm{Sl}_n$ are the ones that stabilize some flag

I am looking for a reference for the above statement that every parabolic subgroup of $\mathrm{Sl}_n(\Bbbk)$ stabilizes some flag in $\Bbbk^n$. I have gone through a large pile of books and can’t seem to find one. Thanks a bunch in advance!

Edit: I understand $G=\mathrm{Sl}_n(\Bbbk)$ as a connected algebraic group and define a parabolic subgroup $P\subseteq G$ to be one that contains a maximal connected solvable subgroup. I know how this is equivalent to $G/P$ being complete (or projective).

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