Cookeh, I was lucky growing up, my parent let me spend all available cash on BMX parts, then still picked me up from hospital whenever I broke something, however as your parents seem so focused on academia, maybe lead with the argument that a faster computer will allow you to organise your thoughts more efficiently while doing uni/college/sixth form work.
point them to this article
http://www.tomshardware.co.uk/hourglass-syndrome-computer-rage-SMASH,news-34735.html and say although you don't suffer from rage
that it can ruin your flow when you have a good idea.
Another angle to approach it from is energy efficiency, as a cheap i3 530/540 build will use about 10-15% of the power of a pentium 4 (is that what you're running atm?) to complete the same workload.
also maybe consider looking second hand, as you've said you can make £100 a day, and for around £100 you could get a core2duo and a g31 board and 2gb ram easily I would think.
(in fact I just priced it up on the site, and brand new, a c2d e3500, MSI g41 board, and 2gb kingston value comes to £107.20 + P&P),
I cant see any logical argument against spending ONE DAYS WAGES on something that will make it easier for you to do the work you need to do to succeed in later life.
Not to say they won't find some illogical argument against it, they are probably afraid that if you had a better computer you'd spend all your time gaming instead of working