Error while creating index in SQL Azure
The catalog view sys.all_objects won't show nonclustered indexes.  Here's the query you should be using:

select
    object_name(object_id) as table_name, 
    name as index_name
from sys.indexes
where name = 'IX_Children_ParentID'


Or to get all of the indexes for that table:

select
    object_name(object_id) as table_name, 
    name as index_name
from sys.indexes
where object_name(object_id) = 'Children'