FYI This has shown it's more than just 'taking the call off my hands' - all the Open Reach call can achieve is getting lines/poles/cabinets/etc. made "safe" and won't help getting service restored!
Openreach are the ones who will fix the fault/restore servce - but they need the Post Office to properly lodge it first - which is effectively all they needed to do.
You should never have been told or needed to call Openreach who don't deal with members of the public - unless there is a safety/health issue involved - hence they making the line safe.
You can't blame the OR engineer for only doing his job, if he had spoken to your parents and fixed their line, he would be breaching the rules (imposed by the regulators) as well as doing something he had not been tasked to do.
Openreach did recently suggest that they would be more willing to deal directly with members of the public rather than just the communication providers (who are their customers) but some of the CP's objected.
They are telling me the ticket is still waiting for OR to reply and has 2 days, 12 hours to run. I asked if that is normal for OR not to have given a date and he said it was and there was nothing that could be done until OR reply.
The response you have got from the Post Office imho is that they are still not escalating your case sufficently as I imagine there are ways they can escalate the fault with Openreach rather than just waiting for a ticket to be updated (perhaps the PO picking up the phone to speak to them etc.)