I have edited in some remarks from the comments ([1](https://mathoverflow.net/questions/385873/parabolics-and-simple-roots-for-a-special-unitary-group-reference-request/385903#comment983612_385903) [2](https://mathoverflow.net/questions/385873/parabolics-and-simple-roots-for-a-special-unitary-group-reference-request/385903#comment983785_385903) [3](https://mathoverflow.net/questions/385873/parabolics-and-simple-roots-for-a-special-unitary-group-reference-request/385903#comment983788_385903)).  To start, I am used to thinking of the stabiliser of a self-dual flag, but of course every self-dual flag has an isotropic flag as its “bottom half”, and every isotropic flag can be completed to a self-dual flag by tossing the duals “on top”.  ([Yu - An introduction to explicit Bruhat–Tits theory](https://web.archive.org/web/20070824032813/http://www.math.purdue.edu/~jyu/notes/banff2001.pdf) is where I first encountered the description of Bruhat–Tits buildings of symplectic groups in this ‘self-dual’ language.)

$\DeclareMathOperator\tr{tr}$Write $L^n = V^* \oplus \overline V \oplus W$ for some $L$-vector space $V$, where $(W, H)$ is anisotropic and $H(v^* \oplus \overline v) = \tr_{L/F} \langle v^*, v\rangle$.  A choice of $L$-basis $(v_i)_{i \in I}$ for $V$ gives rise to a split torus $S$ in $G$, consisting of the transformations that preserve the $F$-line through each basis vector.  Then $S$ is a maximal split torus in $G$, since its centraliser in $G$ is the product of the anisotropic group $\operatorname{SU}(W, H)$ with the torus consisting of the transformations that preserve the $L$-line through each basis vector.

Write $(v_{i^*})_{i \in I}$ for the basis of $V^*$ dual to $(v_i)_{i \in I}$. Put $J = I \cup I^*$.

For each $i \in J$, the map $a_i : S \to \operatorname{GL}_1$ that sends $s \in S$ to $s v_i/v_i$, in the hopefully obvious notation, is a relative root if $W \ne 0$; and its root space consists of all those skew-adjoint endomorphisms of $L^n$ that annihilate all $v_{i'}$ with $i' \ne i, i^*$, and that carry $W$ into $L v_i$.

For each pair $i, j \in J$ such that $i \ne j$, the map $a_{i j} = a_i - a_j$ is a relative root; and its root space is the set of all skew-adjoint endomorphisms of $L^n$ that annihilate all $v_{j'}$ with $j' \ne j, j^*$, and that carry $L v_j$ into $L v_i$.  (Note that $a_{i i^*} = 2a_i$, so, to get the full root *algebra* for $a_i$—if it is a relative root—we need to take the sum of the $a_i$ and $a_{i i^*}$ root *spaces*.)

Since these spaces, together with $\operatorname C_{\mathfrak{su}(L^n, H)}(S)$, span $\mathfrak{su}(L^n, H)$, we have found all relative roots.  The relative root system is of type $\mathsf C_{\dim(V)}$ if $W = 0$ and $\mathsf{BC}_{\dim(V)}$ if $W \ne 0$.  If we identify $I$ with $\{1, \dotsc, m\}$, then one choice of simple roots is the union of $\{a_{i(i + 1)} \mathrel: 1 \le i < m\}$ with $\{a_{m m^*}\}$ if $W = 0$, or with $\{a_m\} if $W \ne 0$.

I will now appeal to the parameterisation of (rational) parabolics by (rational) cocharacters, according to which we attach to a (rational) cocharacter $\lambda$ of $G$ the parabolic $P_G(\lambda) = \{g \in G \mathrel: \text{$\lim_{t \to 0} \lambda(t)g\lambda(t)^{-1}$ exists}\}$.  (This parameterisation has the advantage that it also singles out a Levi component $M_G(\lambda) = \operatorname{Cent}_G(\lambda)$ of $P_G(\lambda)$.)

Fix a rational cocharacter $\lambda$ of $G$.  After replacing it by a $G$-conjugate, we may assume that it takes values in $S$; and, after replacing it by a further Weyl conjugate, we may assume that $\langle a_{i j^*}, \lambda\rangle \ge 0$ for all $i, j \in I$.  Then $P_G(\lambda)$ is the stabiliser of the flag
$$
\Bigl\{\bigoplus_{\substack{j \in I \\ \langle a_{i j}, \lambda\rangle \ge 0}} L v_j \mathrel: i \in I\Bigr\}
$$
(where I've put $a_{ii} = 0$ for each $i \in J$).
 
Since every isotropic flag can be conjugated into $V$, we have shown that the parabolics are precisely the stabilisers of isotropic flags.