$\DeclareMathOperator\tr{tr}$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 maximal split torus $S$ in $G$, consisting of the transformations that preserve the $F$-line through each basis vector. (Proof: the centraliser of $S$ 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 dual basis 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.
Fix a cocharacter $\lambda$ of $S$. After replacing it by a Weyl conjugate, we may assume that $\langle a_{i j^*}, \lambda\rangle \ge 0$ for all $i, j \in I$. Then the parabolic corresponding to $\lambda$ is the stabiliser of the flag $$ \Bigl\{\bigoplus_{\substack{j \in I \\ \langle a_{i j}, \lambda\rangle \le 0}} L v_i \mathrel: i \in I\Bigr\} $$ (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.