After you publish a Learning Story, it will be automatically shared to any parents connected to the child(ren) that you selected. If you would like to keep this hidden however, you can choose to hide the Learning Story by pressing the Options button and choosing Not Share

At which point the Eye Icon on the Learning Story will be turned grey to indicate it is no longer being shared

If you would like to share the Learning Story once more, you can once again press the Options button and choose Share Story

And the Eye Icon will return to the blue state to indicate that it is being shared again