If you are only using the default web context (and if you don’t know, then you are), the following code will tell you if the current user is logged in to the front end:
$isAuthenticated = $modx->user->hasSessionContext('web');
If you are using multiple contexts and want to know if the user is logged in to the current context, you can do this:
$isAuthenticated = $modx->user->hasSessionContext($modx->context->get('key'));
If you are using multiple contexts and want to know if the user is logged in to any of them, this code will do it:
$isAuthenticated = $modx->user->hasSessionContext(array('context1', 'context2', 'context3'));
About the parent field: Inheritance of permissions for user groups is planned for the future but not implemented yet.