I wouldn't call it customary, but whether or not you decide it's best really depends on how "customary" your soon-to-be fiance and father/parents are. Do you think they would react well to, and appreciate, it? Would her parents find it odd for you to "skip that step?" You're probably close enough to the situation to answer those questions.
I have a friend who recently got engaged, and her boyfriend did it the old fashioned way: he went to her father privately before asking. Her dad is so laid-back, it's hard to think of that as necessary, but it certainly didn't hurt. I also have a cousin who got married several years ago, and the guy asked her mom privately (he showed up with a box of Godiva), since her dad is out of the picture. I agree with others in that it's not really about "asking for her hand" anymore, it's just kind of a polite heads up and about getting their blessing. Personally, I would find it more appropriate for the guy to talk to both of my parents rather than just my dad, but I can see how that might be more intimidating.
Bottom line is, I don't really see how talking to the parents and getting their blessing could hurt the situation, unless they dislike you or something. If you all have good relationships, why not?