I second researching things together, try to make them aware of the poor health decisions they're making, and try to help them find alternatives that they'll like.
My boyfriend and I are currently working hard to make some positive lifestyle changes, and we finally got to the point where we were comfortable adding in gym time a few weeks ago. I'll tell you what steps we went through, trying to make it as gradual as possible so that neither of us felt too rushed, or like we were being deprived of anything...
1.) I started cutting out soda, starting with just not having soda at work (beyond a morning Red Bull, which I basically consider a coffee substitute). I got a plastic cup that I refill throughout the day, and I get a TON of water into my body that way (at first, I was hitting around 72 ounces in the 9 hours I'd be at work). I could have soda at home at first, but I found myself not wanting it as much. Now, I just have the occassional soda when we go out to eat, and only if they have diet Mountain Dew or Dr. Pepper. While I hate all the chemicals in the diet sodas, I hate empty liquid calories a *lot* more.
2.) We then started trying to cook more, especially in advance. We turned to fast food a lot of the time because we were very immediately hungry with nothing already prepared, so now on Sundays (usually also out grocery day), we prepare EVERYTHING that will store well for the week, and try to portion it out in advance in tupperware as well. This works for my work lunches, as well, so now I have no excuse to go out and get fast food.
3.) Think about replacing different things you use regularly, like if you eat white bread and whole milk. Whole milk has WAY more calories than you need in a liquid - trust me, I grew up in a house where I know I had a glass of whole milk every day. We looked at soy milk as an alternative, and have found it to be really good. We don't use it for everything, as I think the soy milk tastes a lot like melted vanilla ice cream (bonus!), so we keep around a small amount of 2% milk for cooking sometimes.
We also now try to keep heftier whole grain breads around; if you're going to have a sandwich as a meal, this would be pretty legit as a concern. The more hearty your bread, the less you'll need to fill up. Plus, the fiber helps it be more filling and stick with you. Just be sure to eat slowly so your body has time to send the message that you're full!
4.) Once the quality of food has gotten better, really think about the quantity you're eating. This should be a natural transition, but if it's not, seriously start paying attention to your portions. Spend a few days reading labels and actively measuring how much that serving on the back actually is. In a lot of cases, you'll be surprised (cereal was what blew my mind the most).
What will help to start is just giving yourself half what you'd normally eat. After you're done, and give yourself some time for your food to travel and all, you can make a much better and informed decision about whether to get more. If you start out with a lot of food, it's easy to just mindlessly keep eating and end up *over* eating.
5.) Most important thing: nothing is actually "off limits," just be sure you're eating in an informed manner, and eat it slowly. Don't eat cake every day, or cookies, because I'm sure you aren't actually craving them daily. When you have anything like that, you need to make sure you take the time to savor and enjoy it, and be aware of what a serving actually is!
6.) Get up and move around! You don't have to start big, but maybe start going on family walks after dinner. Try to go out to state parks and explore, walking the trails at whatever pace you're all comfortable with. Do pushups/crunches/squats/etc - you don't need a gym to be active. Know your limits, give yourself time to rest your muscles when strength training, and most of all make sure you actually enjoy what you're doing.
I think that's all I can really come up with to do to help there with your situation. Talk to your parents about making these positive lifestyle changes, taking it slowly and as a family. If you can all work hard and work together, you're more likely to succeed at giving yourselves healthier, longer lives. Even if you don't lose much weight, if you're eating right and staying active you're much better off.
Of all of my tips, I'd say that cooking things in advance and learning portion control can be the trickiest to start, but easy to maintain once you've been actively working at it for a few weeks. Plus, water is highly important, but it's way easier to introduce when you just make every other soda you get water instead, and just go from there!
Good luck, and let us know how things go with your family! It's terrible watching people you love doing such self-destructive things, but certain kinds of food are addictive due to how they affect your brain, and if you're not dedicated to making lasting changes and sticking with them until they become your new habits it just isn't going to work.