I wish someone would put me in that kind of difficult position! But seriously, it's good that you want to be independent but don't look a gift horse in the mouth. As happened in your gap year, you had good intentions and worked hard but still came unstuck because of poor planning and lack of experience, no big deal, and you were lucky to have your parents to help you out. Uni is such hard work, there is no point getting a job for the sake of it. It would be better to concentrate on your studies and get the best degree you possibly can while enjoying yourself and leaving enough time to relax. Also make sure you sort out plenty of work experience etc in the summers. Stuff that will enhance your CV rather than destroy your soul. It really isn't fun being on the brink of your overdraft limit with no one to help you out, and ending up with £20k of loans. And then being so exhausted from 20 hours of lectures and your own study time and then having to work as well.
Maybe see how things go at uni. If after a semester or 2 you feel you can take on a job as well, then go for it. If your dad still wants to give you some allowance on top of your wages then you could be really sensible and start saving. Or if you can find something relevant to your career you can get some unpaid work in that field.
I think it's good that you actually want to be responsible for your actions, but don't resent your parents for wanting to help you. If they can afford it and are willing, maybe you can reach a compromise. Like a monthly allowance that they are not allowed to go over, which will force you to budget and think about everything you need to pay for, and prepare you for real life when you might not have or want the luxury of your parent's support. It's good to have a safety net. As long as you realise the value of money, which to me it seems you do, I wouldnt worry about getting a job just to prove a point.