Are two logically equal indices physically the same index?
How can the 2 indexes be physically the same? The values of id could be more example: 

1,2,3,4,6,7,8,9


while the parent_id values could be:

NULL,1,1,8,NULL,NULL,2,NULL


So the answer to the question is: No, they would be two completely separate and different structures. It doesn't make any other sense. Even if the parent_id were all, one by one, identical to the id values, it still would not make any sense either. Because the server cannot know or be sure that the values will be always identical. Rows can be updated and new rows with different id - parent_id values can be inserted. So two separate indexes have to be used.