Yes, absolutely. It's about respecting what's important to her. And if she is important to you, then the things that are important to her should also be important to you. And it says to me, based on the reason you gave, that she's thinking about her relationship seriously.
That said, this may reveal something about her values. One thing to start asking yourself is whether you and she share fundamental values and judgements. If you do not, you may have long-term difficulty.
From my perspective, I would certainly look more kindly on you as a potential suitor of my daughter if you asked if it was alright to take her out on a date. After you brought me a Silmaril from the Iron Crown of the Dark Lord Morgoth.
*Edit*
My perspective on the "ownership" angle people seem to be taking: It's not about ownership. (Though it may indeed be for some people and some cultures, I don't think it has to be to go along with this.)
It's about my responsibilities as a parent. I have a solemn, serious, nigh sacred duty to protect all of my four children, boys and girls. To nurture and encourage them, and set them on the path to happiness and well-being. That responsibility is largely accomplished while they are my wards, but will extend, in part, until the day I die.
So if my daughter ever sends a man to ask my permission to date her, I expect that she's taking my duty seriously, and wants to see what I think. She knows that, no matter what, I have her best interests and happiness at heart. You, she's not so sure about yet. I would hope that she would take my opinion into account, but certainly don't expect it to be the ultimate word on the matter. (Especially if she were an adult and out of the house.)
As to why you ask the girl's father and not the boy's; the cold, sad fact is that even today, women are in the more vulnerable position if something goes wrong. Women get abused, coerced, and forced on a level that men do not. And, of course, if a pregnancy occurs, it's not the man who has to deal with the most immediate consequences.