Definitely not a proposal in public. It should be a very private romatic moment, perhaps in a beautiful natural setting somewhere - in a wood on a sunny day, on a deserted beach in the moonlight, etc which in later years will be part of your memories. If you want to go down on one knee, that's fine, but only if you feel comfortable about doing it.
As for asking her parents, I think it's still a courteous thing to do, to ask their blessing, even if you plan to marry anyway. It gets your relationship with them off to a better start.
And some advice from a woman - I suggest don't buy the ring first. She might hate it if your taste doesn't coincide with hers, and then she'll be embarrassed about telling you the truth. Make it part of your celebrations to go and choose it together and then you won't get it wrong.
Having said that, I was proposed to on Paddington Station last thing at night just as my last train was about to leave. I barely had time to say Yes before I had to dash down the platform to avoid missing it. And I had to buy my own engagement ring because my husband didn't have a penny to spare. We still laugh about that 40 years later which goes to prove that if you're right for each other, however you propose, everything will turn out OK in the end.