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I had a similar issue - once a month at work, I have a bunch of reports that I have to run. Took me a while to figure this out, but the following seems to work:

Set up the parent with the appropriate start and due dates. Make the parent repeat at whatever timeframe you need.

Add the children under the parent with the due dates as needed, but do NOT make them repeat (otherwise, you get the behavior you noticed where the next iteration reappears in the current parent task).

Also, set the parent up to mark complete when completing last item.

This should give you a task which after checking off will create a new iteration of itself at whatever interval you wanted, and the children will be part of that, set to their respective due dates in relation to the parent's date.

Hope that helps.