I’m not gonna relate the whole story, but here’s some tips based on my experiences:
Make it obvious through your body language and your words that you have every intention of being nice to and are into their daughter, but don’t touch her too much.
Depending on where the meeting’s taking place- offer to pay (if you can, and back off graciously if they say no), or bring a bottle of decent wine if they’re drinkers and you’re over 21. If it’s at their home, compliment the food if there is any. Find something in the decor you like and tell them why. Follow their house rules.
Dress well, but not pretentiously. (an ex once met my dad for the first time at a nice restaurant in flip-flops).
It’s ok if you’re a little nervous. It’s a sign that you’re doing something you consider important. They may be nervous too.
Be the best version of your genuine self, and good luck. :)