Okay, here's where I can speak from experience!
Grab a beer or six and sit down with your Dad to talk about it.
I did that, and my Pops told me that; if I got my sh(nizz)it together, I stood a chance of holding on to the girl. Dad was always heaping praise on me, that way......
Mom, on the other hand, alternating from being slightly less enthusiastic than Dad, to being ecstatic about it (she made Mrs StDr's wedding dress (one of Mom's custom designs) and kept the $750 bill from my future mother-in-law), to crying about "her baby boy growing up and getting married". It was kinda like living in the room next to a couple of bi-polar personlities all wrapped into one Mom.
Just tell Dad that this girl thinks you're her sunup and sundown, and you have no clue why she does; but you (obviously) feel the same way about her and that you want to grow old with her. It might or might not be a good idea to add "Just like you and Mom did" - you know him better than we do, so use your judgement on that one
Once you get Dad on your side, go with him to your Mom. If Heather's Mom is on your side, you should have a slight downhill slope to getting your Mom and her dad to agree, as well. The major effort will
probably be getting her Dad to go along. Dad's are just funny that way about their daughters..... although I sometimes wish I were a teensy little more(okay, a super-tanker-load more) reluctant to go along with Tink's abortive first marriage. (Sorry, daughter of mine; but the truth is finally out)