Changing a parent is implemented via Jira links. When you set TASK-1 as a parent of TASK-2, Gantt Suite creates Jira link between TASK-1 and TASK-2. The link type it takes from the chart configuration. These links are visible on standard Jira Issue details, in Links section.
This functionality may differ from what you might expect for Epics. When you move a task inside an Epic, Gantt Suite creates a link. It does NOT put the task into the Epic like Jira does it. This will eventually be implemented, but not yet available.