Note for others and for future reference: It's not about permissions. Making post-creation changes to sub-project/milestone-ness is not available in the UI. It's a command-line/database edits only thing.
sudo /srv/phab/phabricator/bin/move_project --milestone --child "Responsive-Vector" --parent "Vector" --keep-members child
Note that Milestones (which are NOT considered subprojects) cannot specify members. Instead, the members of its parent project are treated as being its members.
However, the new milestone Vector (Responsive-Vector) under the parent project Vector still displays members in the UI, as before:
The members of the top-level project Vector (Responsive-Vector) were: @Jdlrobson
The watchers of the top-level project Vector (Responsive-Vector) were: @Jay8g
I probably should not have used --keep-members child. Sorry for that.