When it happens, step back and do your own thing. Ask your dad if he wants to go to a ride with you, head to a shop, get a snack.
My mom and I fight all the time and it started when I was about your sister's age (I'm 33 now). The older I get, the more my mom nitpicks every little thing in my life. My defense mechanism as an adult is to not reply - and I mean not AT ALL. Phone conversations, I can sit and stay silent for five full minutes while she goes on a roll. If it's important to me, I'll start arguing with her. For instance, she was insistent that she had never taken me on a cross-country trip and was essentially calling me a liar when I said she took me to
Disneyland at age 3. She was criticizing me for driving cross country with my daughter, who is 2, this summer, even though my daughter has toys and movies and the iPad to keep her entertained. I argued that I had to travel cross country when I was 3 and had none of the amenities that kids today have, but apparently it's an important enough issue for her
now but not when her own daughter was trapped in a car and bored. My dad finally had to step in and correct my mom that, yes, they did take me there when I was a toddler, but my mom is getting very bitter in her old age. I let the subject drop, but she never apologizes about anything like that and after 15 years, it starts to wear you down.
Now, when this behavior was just starting as I was becoming an adult (about 18), I didn't have the experience or maturity to know to "choose my battles" and "do not engage". The nitpicking would start and I would dive in and argue until I was blue in the face. I was also getting high blood pressure and the beginnings of an ulcer, which helped make my "suffer in silence" strategy a little easier.
So I get where you are coming from because I now hate putting my dad in a situation where he has to referee (basically your position), but there's nothing you can do. You're old enough, I would be upfront before you even get to the parks and tell them you will walk away and do your own thing if people are going to argue and try to ruin your trip. Don't be confrontational about it or that will just start a fight, but work it into a conversation that you want this to be a pleasant, fun trip, so if anyone starts fighting, you will excuse yourself and meet up with them later.