Its just over £55 a week.
The thing to think of is what your dad could do if your mum were not there. If he would definitely need external help during the day, he should apply.
If you could somehow get them to apply suggesting this would allow them to pay for somethings around the house that they are now struggling with, you get their consent for both the application and for the help!
The problem is that you have to be quite blunt about your dads problems. I suspect they are in denial about some of his memory issues. I managed to use the ruse that my mum did not have capacity so I was able to sign on her behalf. Even so I struggled to be realistic about her problems, but was rescued by the manager of her sheltered housing who rewrote it for me.
I hate to say it but not taking the help available is pretty typical. Everything rumbles on until there is a crisis. However you need to make sure you are not used to prop up an unsustainable edifice. Pace yourself. I have been involved since my daughter was 10. She is almost 20 now. My mother could live another decade. You are offering a partial solution. They will probably say no, but this gives you a bit of a get out when they call on you to do something for them.
(With Attendance Allowance there is nothing to stop them giving he money to you to help pay for your time, if that is a factor. Actually there is nothing to stop them spending it on chocolate or gin, though either might be of more use to you.)