It's tradition that the man asks for his girlfriend's hand in marriage from her father. But that narrative doesn't necessarily fit with today's couples. It may not be a man who is purposing. It may not be a man and a woman in the first place. And what about families that don't have a father? Or families with two fathers? Also, what about that whole Feminism equal rights thing? In 2017, is the person who is purposing still expected to ask for "permission" first?
Weddings are an expression of your relationship, so the proposal should be too.
What is right for one couple might not be right for another. Is family very important? If so, then you should include them some how. When my then boyfriend/now husband and I were talking about getting married, I felt as though I had to tell my family. I didn't want to catch them off guard. Little did I know, my husband had already spoken to them weeks before. Needless to say, my family tried their hardest not to crack up when I was on the phone with them awkwardly explaining that we may be taking the next step soon. They were in on the surprised, and they really appreciated that.
"Permission" is not the right word. Ask for their blessing.
The feminist/adult in me wants to say, "No! No one gives me permission to do anything. I'm my own person!" And it's true. But at the same time, marriage is about uniting two families. My family calls my husband "son" and his family calls me "daughter." I am really lucky to have such loving, supporting families on both sides. Part of that is that we really wanted to include them. Would you propose to someone without tell your parents first? To me, it's the same thing. I like the word "blessing" because it about support, not ability. You're "allowed" to make your own choices, but asking for someone's support or blessing shows your respect for their thoughts and feelings.
You don't have to ask the father.
Who is your SO most close with? Maybe biological family is less important to them than their friends/adopted family. Maybe they were raised by their older sibling. Maybe their grandparents are very influential in their life. Then those are the people should talk to. Before proposing to me, my husband sat down with my mother, my father, and my little sister. He took them out for coffee, asked for their blessing, and showed them the ring. After, he talked with my uncle/Godfather as well. (My uncle is unmarried and never had children -- he always says I'm like a daughter to him.) My husband made sure to talk with all of the most important people in my life before proposing.
Ultimately, you need to do what's right for both of you.
If you don't know what your SO would like, ask their best friends. My best friend was actually a bit annoyed that she didn't get to know about the proposal before it happened. Their friends will be thankful you talked with them and should be able to steer you in the right direction.