After the rollout and migration of the new "Parent" field, it is editable as long as the user has permission to edit issues.
(This field is also known as "IssueParentAssociation")

Previously, the "Parent Link" field was locked for users to edit from the issue view.

Granting the ability to lock this field (bring back the old field behavior) would be beneficial for some customer implementations of the field.

Update: Currently, the field can be edited even when the Parent field is removed from the edit screen.