If you send the tree in preorder traversal order with known depth, you can send the tree without node ids or parent ids! You can just send the level for each node and recover the tree structure with a stack.
But, indeed, depth vectors are nice and compact. I find them harder to work with most of the time, though, especially since insertions and deletions become O(n), compared to parent vector O(1).
That said, I do often normalize my parent vectors into dfpo order at API boundaries, since a well-defined order makes certain operations, like finding leaf siblings, much nicer.
I think you can still have the functionality described in the article: you would send “hole” markers tagged with their level. Then, you could make additional requests when you encounter these markers during the recovery phase, possibly with buffering of holes. It becomes a sort of hybrid DFS/BFS approach where you send as much tree structure at a time as you want.