|Hit CTRL-D to duplicate the Sphere. Open it's attributes menu and change it's name to something like "Clouds". While the attributes menu is open, adjust the object's size so that it is one unit larger than before on all axes. Also, click the Linking tab and set the object's parent to the previous sphere. Once you have set the parent/child relationship, you can move and resize the whole group through the master sphere (Core Planet here). In Bryce 2 you will want to group the objects when you are done.
Repeat the process for as many cloud layers as you want, the new objects will share the same parent.