Thanks Klas again for the prompt response and clear answer.
The problem is that it's impossible to use art_order field both in parent and child on same page I guess.
I've tried to make a separate drag-drop-redorder search-list for the parent-list. I jump to this new search-list by button-free on the parent the returning again to the parent-child-search-list (with refresh). This works fine the parents appear in a re-ordere way.... but.... the child-list won't save the ordering when I try to re-order them...
For now it's no problem because drag-drop-reordering by art_order is not needed for the parent. I'll order the parent on date by default.