$\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\SO{SO}\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\PSO{PSO}$Irreducible complex flag manifolds are of the form $X=G/P=G_u/K$ where $G$ is a complex simple Lie group and $P$ is a parabolic subgroup, and $G_u$ is a compact
real form of $G$ and $K=P\cap G_u$. As a rule, the connnected automorphism group $\Aut^0(X)$ coincides with $G$, with the only exceptions:

1. $X=G_2/U_2$, $\Aut^0(X)=\SO_7(\mathbb C)$.
2. $X=\Sp_r/\Sp_{r-1}U_1$, $\Aut^0(X)=\PSL_{2r}(\mathbb C)$.
3. $X=\SO_{2r+1}/U_r$, $\Aut^0(X)=\PSO_{2r+2}(\mathbb C)$.

You can find a discussion in chapters 3 and 4 in [Lie group actions in complex analysis](https://doi.org/10.1007/978-3-322-80267-5), by Dmitri N. Akhiezer, Aspects of Mathematics, vol. E27, Friedr. Vieweg, Braunschweig and Wiesbaden, 1995. 

**Edit**: Thanks to [abx](https://mathoverflow.net/a/160305) for the further explanations and original papers.
I just wanted to point out that the reducible case and the full automorphism group are also discussed in the book by Akhiezer. In the reducible case, 
the connected automorphism group splits as the direct product of the connected automorphism groups of the factors. Moreover $\Aut(X)$ is the semidirect product
of $\Aut^0(X)$ by a finite group isomorphic to the subgroup of the automorphism group of the Dynkin diagram of the complex semisimple Lie group $\Aut^0(X)$ preserving the vertices corresponding to the parabolic subalgebra.