Assuming an authentication token has been generated for the parent account, Single Sign On can be set up with ease. Since the parent account is already authenticated, specific login URL's can be generated for the child users using the Authenticate Child User call. Simply pass the username of the user you wish to authenticate as well as the authentication token of the parent account (like the one you can generate above) to retrieve the data.
The URL looks like this:
Once the response comes back, you'll want to grab the JSON object loginURL.
Note: Do not include the curly braces with your username or authentication token.