Create your parent object, then create one child. copy the array from the parent to the child Ctrl-L. next take the child and add drivers to all of the parameters in the array modifier that you want to auto update. now copy the drivers and array modifiers onto all of the other children. Done!
Using drivers is tricky, Here are some tutorials:
https://www.youtube.com/watch?v=o4qYHNA_x4Q