Parent issue summary is not displaying in Child issue under Issue Links module.
True, I don't see the parent issue listed in the issue link when looking at the subtask. However I would not have an expectation that this would happen though. The relationship between a parent and a subtask is not the same as an issue link in Jira. Issue links can be linked to any other issue in Jira regardless of project.
But a subtask can only exist when it has a reference to a parent issue. That parent issue has to be in the same project as the subtask. These are not really issue links though in Jira. Yes there is a relationship between these issues, but it's a different kind of relationship than that of other issue links.
For an example, here is a parent issue I have in Jira that has 2 subtasks, but is also linked to another issue in a different project:
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