Don't DND settings use local time? So would not be active at the same world time, but would automatically be active on the same local-time schedule, depending on where the iPhone is actually located? (i.e., Your relative lives in Australia, then comes to New York, and would not need to adjust DND settings, as the iPhone knows what time it is locally, so DND would be at the same (local) time, no need to adjust for that, the iPhone takes care of it...)
Not sure if I understand - your parent has an iCloud account (assuming an AppleID, so they apparently have an iPhone, but you want to keep an iPhone that is always set up for their iCloud account... (So they don't have to bring their own iPhone???)
I don't want to know family "stuff", but just wondering why you need to keep a phone, set up just for your parents?