It gets complicated to compare the different ways to parametrize or realize a unipotent class (or equivalently, in good characteristic, a nilpotent orbit in the Lie algebra). But I think the answer to the basic question here is no, unless I'm misreading it. When a class happens to be Richardson (the unique orbit intersecting densely the unipotent radical of a given parabolic subgroup), the dimension of the class is twice the dimension of the unipotent radical in question. The class can also be determined by the Bala-Carter method, starting with a Levi subgroup of $G$ and its Borel subgroup or other distinguished parabolic. Here the class is the Richardson class determined by that distinguished parabolic.
Taking just the example $D_4$ with the given class being one of two determined by the partition $(4,4)$, either class (and a third one as well) has dimension 20. Here the Richardson viewpoint starts with a parabolic subgroup of Levi type $A_2$ whose unipotent radical has dimension 10. But the Bala-Carter viewpoint starts with a Levi subgroup of type $A_3$ together with its distinguished parabolic (Borel) subgroup having a unipotent radical of dimension 10. So the two Levi subgroups in the picture are far from being conjugate.
Generally speaking, the Richardson method starts with a big parabolic subgroup (and small Levi subgroup thereof) to yield a big class, while the Bala-Carter method starts with a big Levi subgroup to yield the same big class. The former method doesn't usually yield all unipotent classes, whereas the latter method gets them all. But the two methods are roughly dual.
Going back to $D_4$, what you can observe in the closure diagram is a duality between the two situations in which a single partition labels two classes. But there is still the notational task of sorting out which parabolic or Levi goes with which of the two classes in each case. There is a lot to be said here in general for type $D_{2n}$. Let me just add that unipotent classes or nilpotent orbits don't depend on the isogeny type of the group, so special orthogonal groups are convenient to work with.