Wel if you are big into gaming, neither the mac book pro or the laptop you linked have a video card, so both of your choices are bad for video games (unless you get the 15 in mac book pro.) Ill expand more on the mac book vs pcs as far as gaming goes later
As far as graphic design, I have used both, and I woukd definitely gives macs a slight edge in this regard, if you use adobe products for most of your grapbic design, than the edge is pretty much lost, since adobe products are available on both mac and pc.
As far as battery life macs book pros do indeed have pretty great battery life. Its hard to compare that to all PCs thougn, since there are many companies which manifacture pcs. I have had great battery life with Asus computers. I would stay away frol dells and hps. However, remember, if its a gaming pc laptop, the battery life will probably be less than a mac.
As far as word processors, microsoft word continues to be the best. Luckily its available on both platforms. For macs, you will probably have to buy microsoft word. On pc, i believe you can get office at a discount price kr even free depending on what edition of windows you get (ie home edition vs professional edition, etc.)
As far as lasting, I havent had a lot of good luck with mac books lasting me, but thats just me. I have had great luck with Asus comouters though. I had an asus g1 (asus first gen gaming laptop) and it lasted me almost 6 years, and even worked after it had gotten beaten up so badly over the years that the screen was falling kff the base.
Now, as far as gaming goes, you CAN game on a mac. Even windows games, by using bootcamp. Bootcamp basically just installs another operating system (most commonly some version of windows) on your mac. It works amd is perfectly safe, but you will have to buy windows, unless you are ok with pirating it.
Pcs, obviously, dont have any of the above problems. As far as gaming goes, there really arent any drawbacks to having a pc.
However, when talking about gaming, you have to mention performance, and with that discussion yiu must also consider price. You can get a very good gaming laptop for around 1100-1300. This will play most games at high or max settings. For a mac book pro (with the ret display) you will need to get one with a dedicated video game if you want to do any serious gaming. Only the 15inch model has a dedicated video card (a fairly nice one) and the non retina display one costs 1700 while the one with retina display costs 2100. what is most important to consider is the power of the mac book and the gaming pc. lets compare the 15 inc macbook pro (with ret display) to a 1400 dollar Asus gaming laptop (g75vw model. See source for newegg link to it)
The asus laptop is 700 dollars cheaper (it costs 33% less). the mac laptops processer is only .1 ghz faster (a negligible amount, especially since they both have i7s.) The asus has a 17.4 inch screen while the mac has a 15 inch. The asus has 12gb or ram vs. 8 on the mac (which is quite a substantial increase) the asus has a far superior video card (a 670 with 3gb of video ram vs a 650 witg 1gb on the mac) I believe the macbook has a larger hard drive, but the asus has 500 gb, which is plenty of space for a laptop.
Overall you get way more power for way less money with the asus gaming laptop, and it will be able to handle most if not all of your graphic design and word processing needs. If you are a big time gamer, I really see no reason for you to get a mac over a pc (especially if you were planning on getting the asus laptop I linked.) If that asus is too expensive, you can look up the g55vw model, which is only a little bit less powerful than the other asus (still a bit more powerful than the mac book pro 15inch with retina display) and only costs 1000 on newegg.com which is less than half the price! However, a mac book pro will undoubtably have superior battery life.
If you really really think you want the edge that macs have over pcs as far as design goes, amd gaming isnt all that important to you, than a mac may be the way to go. But let me stress that, from my experience graphic design on a mac isn't THAT much better than on pc. Its a slight edge that macs have (again this is my opinion). Also, remember that the battery life on a mac wilk be better than a gaming pc. If battery life is extremely (and I mean, like EXTREMELY) important, than this may be a deciding factor for yiu.
Overall, if I were you, I would ask for the much cheaper and more powerful asus gaming laptop. If your parents really want to soend 2100, you can get a good asus gaming laptop and get a decent gaming desktop as well. All for the price of one mac book. Now, price might not matter to you at all, in which case the decision is harder to make, but you could spend 2000 dollars and get a top of the line PC thats WAY more powerful than the mac you coukd get for the same price.