I'm asked to find all the parabolic groups $P$ which contains $T_4$, the subgroup (borel) of upper triangular invertible matrices. This is my definition of parabolic subgroup
Let $P$ be a subgroup of $G$, then $P$ is parabolic iff $G/P$ is a projective variety.
Over the internet ( for example here) I found another definition of parabolic group of $GL_n(k)$ which is:
$P \leq GL_n(k)$ is parabolic if stabilises a flag in $k^n$.
I'm not so practical with this notion, can someone explain me the equivalence of this two definitions?
I might be mistaken, but I think $k$ at here is assumed to be algebraically closed. Then $G/P$ is a projective variety implies that $G/P$ is a complete variety, whereas later is equivalent to $P$ is any subgroup between $G$ and $B$, since $B$ is the minimal subgroup with this property. Note that for semisimple algebraic groups, $G/B$ is in fact a flag variety, so we cannot get any smaller ones. The part that $G\supseteq P\supseteq B$ implies $P$ acts as a stablier for a certain flag is proved in the file you linked.
Also this one seems largely a duplicate of this question:
Parabolic subgroups of $\mathrm{Sl}_n$ are the ones that stabilize some flag