The study lumps red meat and processed meat together. You can't do that. Eating proper portions of lean red meat is very different from eating hot dogs every day. The study also doesn't differentiate between lean red meat and fatty red meat. Very poor science in my opinion. It's almost as if the study was molded to fit a pre-determined conclusion. I would be curious to see the motives behind the study. I'm guessing PETA or vegan roots.

The greater battle is getting them to understand not just that this study is incorrect but that they all are. The best ammo is Good Calories, Bad Calories by Gary Taubes. It goes through many of the studies and the history of the 'science' that claims saturated fat or even fat in general is bad. If you haven't read it, check it out. It's dense but I'd say if you are off reading Tale of Two City's in High School then this book should be easy and far more interesting.

Bonus if you can get your Dad and Mom to read it. It will help them see your perspective and perhaps save their lives.

Taubes has an undergrad in physics and applied engineering and his early science writing career focused on physics. He's a smart guy who spent the entire book analyzing and debunking studies like the one your dad sent you. He has won an science journalism award for his article critiquing the demonizing of salt in diet as well. http://en.wikipedia.org/wiki/Gary_Taubes

Here is a link to the book on Amazon - http://www.amazon.com/Good-Calories-Bad-Controversial-Science/dp/1400033462/ref=sr_1_1?ie=UTF8&s=books&qid=1279897754&sr=8-1

The study not only lumps red meat with processed meat and does not consider confounders like carb intake, but it also found a slight beneficial effect for white meat. My advice, tell your parents that just to be safe, you will eat more white meat since it has a protective effect. Then be sure to eat more white meat IN THEIR PRESENCE. ;-) You might also want to dig up better quality studies that show that eating meat is good for blood lipids, etc. There are tons of these in the paleo community and on paleo blogs. Also, what works really well for me is to tell them that you have cut out all nonhealthy foods like noodles and sugar and cake and cookies and soda. Therefore, the end result is you eat more meat, veggies, and fruit. Tell them the meat is healthier than the cookies and noodles.

What I do in general is tell people I have cut out all processed sugars and starches because they are bad for the blood and body weight and they contain very little nutrients. "Low carb" has a negative connotation so I just don't mention that word if I want people to listen. I just say I'm cutting out processed food. Everyone likes the sound of that and I don't get any arguments. If they then give me some story about whole grain, I point out to them that almost all whole grain is made with a lot of white flour with very little actual whole grain and very little actual nutrients. This gets people hooked in and I only speak further if I think they are open to it. Otherwise, I stick with the spin that goes down the best.
-Eva

Simply finding that within a given population those who eat meat get more cancer is meaningless given the number of confounding variables. People who eat more meat are also poorer, less educated, more likely to be male, drink more, smoke more and so on. Such studies also show strong links between eating more red meat and risk of death from driving accidents! That's not to show that red meat makes you a more dangerous driver or makes you addicted to cigarettes...

Edit:

Actually Dr Eades has critiqued the very article which you linked to, and he also links to his two other writings about why observational studies can be highly misleading:

It should be obvious that the shaving or lack thereof has nothing to do with heart disease or early death; the lack of shaving is merely a marker for all the other conditions that are risk factors for heart disease, i.e., small stature, unmarried, smoking, lower socioeconomic class, etc.

How can you respond? Instead of trying to persuade them with a clever answer or counter-study, consider just not responding. The proof is in the pudding (couldn't think of a paleo food to replace pudding...), so show them good results in terms of weight loss (if relevant), health, vitality, etc. By your deeds shall you be known.