Edit button not displayed, although user has permission to edit child nodes (but not to create)

I am trying to find the right setup for following usecase:

Authenticated users (or other roles) are only allowed to edit child nodes, but not to create new or delete existing child nodes.
So I checked the permissions on people/permissions :
Allowed to see add and edit buttons checked
Certain Content Type (Child nodes): Edit any Content checked (all other edit own, create and delete not checked)

If a user with this role go to node add/edit parent form, both buttons (edit and add) are not shown, although the user has permission to edit child nodes.