I am so sorry for your loss. My dad died more that 30 years ago now, but it still feels like it was yesterday sometimes. I looked after my mum after that and lost her this time last year. Like you, I found myself looking for somewhere to talk and found this site only recently. Like you, I felt a weight of responsibility settle on my shoulders after dad died. I still miss him terribly and now mum too. I still talk to them both daily.
Believe me, five months is nothing and it will be a long time before you feel like it is worth trying to make an effort to keep going on every day and normal is now forever different to what it was. I still have a sense that it doesn't matter what happens to me now. It's like I've served my purpose, but don't forget, both your dad and your mum put years of their lives and lots of love into raising you and I am sure they both feel that your life is valuable. You owe it to them and to yourself not to give up on yourself or your future. Your dad would not want that. Besides, he needs to be remembered in honour of his life and his worth and who better to do that than you, who love him so much. So please don't give up. I am sure he would want you to go on with your life and make him proud and happy that he has such a caring and loving daughter.
Don't be too hard on your mum either. She is no doubt feeling just as lost and lonely as you and perhaps this friendship is a way of filling some of the void and finding reassurance that someone still cares about her. Everyone needs to feel that and it doesn't mean she's forgotten your dad. I expect she is just trying to find a way forward in this new reality, as you are. She probably craves support and friendship and companionship and sometimes it isn't the same and isn't enough to get that only from her children. It's a very different kind of relationship and she probably feels she needs to be there to support you through this and is glad to have a friend who will support her.
Be patient with her and with yourself, but never give up on life. It will get better and your dad, your siblings and your mum will all want the best for you. It is the hardest thing to lose a parent or anyone very close to you, but try to focus on something else, even if only for a few hours a week, to help you connect with life again and to give you a little relief from what you are going through. A year after losing my mum, that is still rather an alien feeling, but I feel it is necessary. You probably won't be ready to do it yet, but you need to try to reconnect with things that give you pleasure in life again at some point, otherwise the grief can be overwhelming. It's something you need to try to find a way to survive and learn to live with and over time, you will. Be patient and kind to yourself. Do whatever helps you to feel better, cry, visit the grave, take flowers, have a chat to your dad, tell him you love and miss him. I am sure he loves and misses you too. Put some favourite photos in an album, plant a rose bush or get a bench made - anything to honour his memory, but be sure he would never want you to give up. He will have felt the same when he lost his parents, but he didn't, so make him proud and keep on going. After a while, you will feel he is still at your side and walking with you and smiling down on you.
Take care and be good to yourself. It is a dark tunnel but light does glimmer at the end eventually.