It's very hard OP, when the main carer is being obstructive, and they sound like very private people. Is your mum able to express a preference? As in, can she tell your dad to bugger off for a walk and leave you and her together for a while? One of our biggest problems was / is that FIL has POA for MIL but he's really not up to it.
It could be that it is so new they are just in 'we'll do whatever it takes' mode. Guilt, and desperately wanting things to go on as 'normal' for as long as possible. As a DIL it was easier for me to step back and see the big picture, but FIL, DH and SIL were just over their heads in guilt and feeling bad for MIL at the start.
They struggled on at home, with increasing help from SIL, until it became clear that FIL could no longer meet MILs need and could not care for her safely. She was very dehydrated (he didn't make sure she was drinking), hadn't been taking her meds (he just handed them to her and expected her to take them, she just stuffed them down the sofa cushions, wrapped them in hankies and stashed them in various places). She wasn't clean - he didn't supervise her in the shower or in the toilet. It came to a head when she became delirious due to a severe UTI and was taken to hospital. That was the crisis that tipped them over: she went into a nursing home straight from hospital. It turned out she'd been having frequent falls, which FIL wasn't telling us about. It was getting to the point that a pp mentions - if she hadn't got ill enough to be admitted to hospital, we would have had to step in, in some way, as MIL was not safe at home with FIL.
So not very cheery but experience from this and from this board is that it'll take a crisis to force a change. Might be a wee one, might be a big one - but they will likely attempt to carry on 'as normal' for as long as they can, will resist all forms of help / change / care from outside, until they really can't - or the decision is taken out of their hands.