You should be fine. I work at a credit union, and have worked in banking for 3 years, and I have seen people in much worse situations that you! I would suggest trying to borrow enough to cover the negative balance from your mom or dad. It sounds like you have your savings linked to your checking for overdraft protection, and that is good. If you are over 18, ask your bank if you can get a line of credit to use as ODP. If you are under 18, you probably won't be able to get a line of credit to use as ODP, although I would suggest doing that once you turn 18. If you do get the line of credit, DO NOT use it for anything else. My old bank gave you an actual credit card- if this is what your bank does, shred the card. Do not get tempted to use the credit for other stuff, because that defeats the purpose.
The only time a negative balance will really hurt you (long term, not just in fees) is if you leave the account in the negative and the bank charges it off (closes it with a negative balance). In that situation, most banks will then report you to Chexsystems or Experian as owing them money. This is bad, because nearly all financial institutions will pull one of the two records before opening a new account. They see people with a chexsystems record as being higher risk than someone without, and usually won't open an account. Also, that will stay on the record for anywhere from 3 to 7 years, meaning a mistake you made at 18 may still affect you when you are 25. I am not trying to scare you- most institutions will give you about a month to get your account caught up before charging it off.
Also, almost every young person overdrafts at some point- it is an expensive way to learn that keeping your check register up to date is very important. I have only had 1 overdraft due to spending in the 7 years that I have had a checking account. I had 2 due to not paying attention when setting up my bills to pay online, and taking them out of the wrong account. I think you will find that after this, you will be much more attentive to your account. Good luck with your dad whenever he finds out!