At your age, your parents are not only thinking about safety, they're thinking about the fact that it will cost THEM more money than it's worth to keep it on the road for you. It's one thing to keep these cars going reliably by doing the work yourself, it's a whole 'nother deal to be paying someone to do this or that to keep it working right. Honestly, old Subarus can be cheap enough that if you (or your folks) get a newer car you could still get the oldie to work on as a "project" that runs and drives. You'd learn a lot too. That said, not only do you need to buy the car, you also need to invest in a decent tool set to do the work on it. The skills and tools, however, you will own for LIFE.
As for safety, yea, newer cars are "safer". The problem is you can then have faulty Takata airbags, wonky gas pedals, bad seat belts, etc. etc. There is a trade-off too. All this stuff requires a run to the dealer or shop if not recalled and that means $$$$.