Just the people who don’t qualify for free flu vaccine will have to pay at Pharmacies. You would normally qualify for being your Dad’s carer, and still get it.
I’ve looked at criteria in Boots chemist website for who get it free, though it’s not very helpful, for me. I have been getting it free for years (as I had a stoke), but it doesn’t actually mention stroke as a criteria for free flu vaccine!
I’ll just wait until I get a text message for my GP, when they are in stock of vaccine & clinics organised. That’s the usual way anyway.