Your parents seem to talk sense Felix.
They only want the best for you. If and its a big IF you get through all your training, PPL, Night Rating, ATPL groundschool, CPL, ME and IR with first time passes, even the cheapest route will cost in the region of £30K. Thats a hell of a lot of money, if you go to a intergrated school it will be double plus living expenses.
SO you get your fATPL and what does it get you...... well the fact is you are not the only one out there, there are many more like you, what makes you more employable? Maybe a degree would help you put your CV up the pile a bit.
The sad fact is that more and more wannabe's are having to pay for a Type Rating and maybe line training as well cost of this can be annother £30K.
So you go intergrated, what does that get you.... well I was at BA, Cranebank a few weeks ago talking to a Training Capt and he was telling me that the two students with him, doing their LST's, were paying BA for their 737 rating. These lads were ex Oxford students and now have jobs with BA.
I myself was offered a job on the condition that I paid for my own rating, which I am doing.
You are young, god what I would give to be 17 again, I would get your PPL while you are at Uni, work dammed hard get a good degree and try to get a good job, then you can always do your advanced training part time.
Good Luck
Tony