I was in a similar situation: My parents paid for the part of my schooling not covered by scholarships (~50%), and I worked summer jobs and internships. I basically was only covering any eating out and books from my own savings. (I didn't have a car or phone.)
With my parent's blessings, since they were my backup plan as well, I used part of that money to start and fully fund a Roth IRA. My plan was to pull out the contribution part if something catastrophic happened and I needed the few thousand dollars I had tucked away. I never needed to, and now those contributions have been growing and earning interest for 10+ years.
If the money you are earning is reported as earned income to the IRS, I would look into doing the same- opening a Roth IRA. (I'm a Vanguard fan, fwiw.) Tuck that money away in a good stock index fund. Yes, Roth IRA is post-tax money, but I bet you're paying (at least close to) no tax now anyway.
I would say to get your parents' blessing before tying up money too much, since you are thinking that they'll be there if needed. It's what I would consider a politeness thing- don't just take that support for granted as you head into adulthood. You may also want to make a savings fund to hold as a 'adult launch' fund for after graduation. Professional/ interviewing clothes, relocating, apartment fees, furnishings, etc. can be unexpected expenses when starting out post school.