I don't know very much about spin groups, but I need to do some reasonably explicit hand calculations in the spin groups over finite fields of odd characteristic in even dimension (the Schur covers of both $D_n(q)$ and ${}^2D_n(q)$ with $q=p^n$, $p$ odd). I'd prefer a representation where the maximal unipotent subgroup, the center, and a generating set of outer automorphisms are very explicit. If parabolic subgroups and their unipotent radicals can also be clear, that'd be ideal.
Is there a reference that works all of these out in at least one representation?
I'd like things to be as clear as SL (upper unitriangular matrices, scalar matrices of determinant 1, {diagonal, frobenius, inverse transpose }, block upper and block uni-triangular).
The matrix representations of size $2^m$ are pretty large and inductive, so I suspect won't be very good. The Clifford algebra seems ok, but I don't actually see any of the pieces there.
maybe this can help? http://www-math.mit.edu/~dav/symplectic_parabolic.pdf it is from MIT seminar 18.704:The Classical Groups and Geometric Algebra
For example, they show $|Sp(2n, \mathbb{F}_q)| = q^{n^2} \prod_{j=1}^n (q^{2j} -1)$