On the phone again to Telstra this morning, and I had some moderate successes but overall what I was trying to do was a dismal failure. I think the people at the Telstra shop see my parents coming!
It turns out Mum's mobile is still under contract, but I expected that. They even refunded part of the recontracting fee, but the ensuing conversation was enlightening. Everything else is off-contract, for now. If we want to change anything, anything at all beyond just continuing to pay the bill, we must sign another contract (that's where the original contract came from, unexpectedly).
That sucks, but basically whatever way we slice it there's no cutting down on the home broadband bill until they can move to a place that has more options available.