I am so very sorry for your loss, I have no idea how hard it must be for you to have lost not one parent, but both, my heart goes out to you.
I am proud that you have reached out here to get some comfort and some support and to chat and to get these feelings off your chest. I can relate somewhat to what you have shared about guilt and about doing things differently, I too have these feelings and have worked hard to come to a better place now with these regrets. I think some counselling would be a wonderful idea, it can really help to put these feelings into some sort of order and process if you like. I learnt so much about grief and about how to work my way through this process, as it is just that, a process and there is no right or wrong way to do it, there is no time frame and there is no expectations. You do it how you need to for as long as you need to.
My loss was through suicide so I also was able to get some support about that too, to learn about suicide which also helped me to grieve. I am wondering if you can perhaps take some time to talk to the doctor of your father to understand more of his illness and this may release you of some of the guilt that you are carrying. Of course hindsight is a most wonderful thing and if we all knew that we were never going to be seeing this person again we would do things differently, however we don't have this luxury in life. I am sure, that like you, each day with your parents you did the best you could with what that day brought to you.
Three weeks ago is not a very long time and I am sure that you are still very raw with emotion from the loss of your father and this has also brought up all the feelings of the loss of your mother so it is a really heavy and emotional time for you Til. Please try to take some time to be kind to yourself, to do something that makes you feel good, you have done nothing wrong here and do not need to punish yourself.
As you mentioned also, you were a young person and could not have possibly taken on the role of caring for your parents so the best solution for them to be cared for was to have them in a nursing home. Maybe in this time some counselling might have helped you process this, but once again, hindsight is a wonderful thing.
I too would be in denial about my parents health, it is a very hard thing I am sure to have to realize that your parents are not well and in palliative care.
I hope to chat to you some more Til, we care so much about you.