For an algebraic group $G$ over an algebraically closed field $k$, a parabolic subgroup $P$ has factorization $P = Q \rtimes L$, where $Q$ is the unipotent radical of $P$ and $L$ is some Levi factor of $P$.

If $G = GL(V)$ with $\dim V = n$, then $$P = \{ \pmatrix{A_{n_1} & & & * \\ & A_{n_2} & & \\ & & \ddots & \\ 0 & & & A_{n_t} } \}$$

for some $n = n_1 + \cdots + n_t$ and $$Q = \{ \pmatrix{I_{n_1} & & & * \\ & I_{n_2} & & \\ & & \ddots & \\ 0 & & & I_{n_t} } \}$$ and for example $$L = \{ \pmatrix{A_{n_1} & & & 0 \\ & A_{n_2} & & \\ & & \ddots & \\ 0 & & & A_{n_t} } \}$$

So $L \cong GL_{n_1} \times GL_{n_2} \times \cdots \times GL_{n_t}$. What about when $G = Sp(V)$ or $G = SO(V)$ in characteristic $p \neq 2$? Then $P$ is a stabilizer of a flag of totally singular subspaces, how does one describe $Q$ and $L$? Is there a good reference for this?