If your computer costs below $300 you could probably get a similar pc for less. Anything above you cannot get one cheaper unless it is on sale. The labor, branding and work costs just make pre-built pc's expensive and a bad option. (additionally they usually have silly configurations which are designed to make the most money for the seller).
Ask anyone who knows anything about computer hardware and they will say pretty much the same.
Here is an improved build for less money but more performance. (Note: I removed the SSD as it is a luxury at this price range and the money could be used much more effectively):
http://pcpartpicker.com/p/KmGq99
I noticed you were missing an optical drive and Operating system in your original build, I was not sure if this was because you had them already or just forgot so here is a build including those two components in case you forgot them:
http://pcpartpicker.com/p/J6htsY
To aid in your endeavor to convince your parents, here is a build which has a similar price but is severely less powerful:
http://www.bestbuy.com/site/cyberpowerpc-gamer-ultra-desktop-amd-fx-series-8gb-memory-1tb-hard-drive-black-red/9177174.p?id=1218994550644&skuId=9177174
And before they pull the 'Oh but it has a warranty' excuse, all the parts you buy usually come with individual warranties that last much longer than 1 year.
There really is no argument, If you want a pc that is more powerful but a lot cheaper, you build it yourself. Pre-built computers are for people who don't have time (a few hours) or just don't know any better.
I hope this helps you get the best PC for the best price.