Sorry you’re going through this.
I know absolutely nothing about UTC but employment wise I do.
Just be mindful that your employer will only allow this sickness to continue for a certain period and then they may dismiss you on the grounds of capability. They need medical information to do that (eg GP report or Occ Health report) so don’t be bullied by them to come back to work or to resign under duress; and actually by being “dismissed” may help you a little with the benefit side. Plus you’d get notice pay and accrued holiday pay (whilst waiting for this to happen you could submit a request for your statutory holiday to be paid to you (legal entitlement) so that might help you out financially.
The other side to this is if work is one of the main causes of the stress, would you consider just resigning? Take the accrued holiday and leave. Sometimes these things never make sense to do, but the feeling of control of that particular aspect in you life might be what you need to do. You may feel a big weight off your shoulders.
This then leads to your plan for the next 6 months. What in your heart of hearts do you want to be able to do? If money and circumstances were no object? If you feel moving in with your parents to help will help, then do it. Don’t delay that decision. If it’s terminal, the time is precious.
I never really understood a friend who upped and moved to the other side of the country to care for her dad - left her husband for 5 months to deal with the house etc but ultimately she knew it was always temporary and her husband would be there forever and she said to me she had to do it - “he’s my dad” was all that I needed to hear to know she was ok and needed to do it. It was tough on the family situation, of course it was, but she hasn’t regretted the decision she made and I know she mentally and physically supported her dad during his last months.
Finall, for your own mental health and well being, what are you doing? I use the analogy that you shouldn’t “fit someone’s else’s mask before fitting your own” (from the aeroplane safety talk) and this is one of those times your needs are also important - you are no good to anyone if you’re not well yourself. If you can ensure good diet, some exercise (stretching/yoga in a morning) & mindfulness techniques (there are great apps out there) - you’ll be a better position to care for your dad (and mum) during this difficult time.
Wishing you all the best.