I apologise if I am repeating what other people have already said and also apologies for any mistakes. I am typing this quickly because I have to be somewhere.
I am going to give you an honest view of Care. We had to have a really big care package for my Mum in the end and dealing with carers and care agencies was one of the most stressful things I have ever had to deal with and it is hard when on top of that you are trying to cope with knowing that your lovely parents are very sick. I am telling you this because you and your siblings will really need to support one another.
With my Mum, we had some incredible carers, who were lovely, funny, kind and a joy to have in the house but we also had some who were not. Dealing with care agencies can be beyond frustrating so you need to do your research.
Firstly, unless you need to get the council involved for financial reasons, don't because it is just another layer of people you maybe don't need to interact with at this stage. Councils will have preferred care agencies that are not necessarily the best. It does sound like at the moment financially your parents can cover the costs.
Do some research using the Care Quality Commission website. All care agencies have to be registered with CQC and are given a ranking. On the website search for care agencies in your area and then look at the ranking and it will give a report about the agency.
https://www.cqc.org.uk/
Also, ask on the local Facebook groups in your parent's area because they will give you honest feedback on which agencies are good and which agencies are not. Also, I apologise, I can't remember the name and probably there is more than one, but there is a website where you can look for private carers, which is cheaper but you run the risk of them not be able to attend if they are sick or there will be no cover if they are on holiday.
An important thing to know is exactly what you want them to do. I am not sure what @Netaporter experience was but from our experience, my Mum's carers did some shopping, did her washing, went to hospital appointments with her, collected prescriptions, cooked meals from scratch, cleaned downstairs in the room my Mum was sleeping in. On top of the personal care elements to looking after my Mum that were required. That however was when my Mum's package was full time.
It depends on how you want the care structured. Do you want a daily visit for a couple of hours where they do a bit of cleaning, and cook some food? Or do you want more visits but for a shorter duration? For example, initially, my Mum had 4 30-minute visits a day where they purely did personal care for her. However, I remember the carers telling me they had a client where all they did was attend in the evening, cook his meal and clear up the kitchen.
Or do you need someone to pop in a couple of times a week, help with the shopping, cleaning, meal prep?
There are options but one of the biggest pieces of advice I can give you is to know and clearly and write down in detail the care plan. This is to ensure that you, your parents and any other members of your family who are involved do not end up having things continually added to your list of things to do.
I will give you another example, the first care agency we used with my Mum after their visits, I would end up having to clear up after them. A lot was going on in my life at a time and this became more for me to do. When we moved care agencies I was more experienced and was very specific about what I wanted from each care visit. I also got incredibly organised, everything had a place, everything was labelled, there was a meal plan, they could add to a shopping list, so every carer knew what they needed to do and everything was always in the right place so they didn't waste time and could get on with their jobs.
I can't stress it enough the more organised you are at this stage, down-to-the-minute detail, the easier it will make it for everyone and will help you transition as their care requirements increase.
I am wishing you good luck and hope it all goes well.