Should I amend my return and start a new one if I did not check the "You as a dependent" box next to "Someone can claim" in the "Standard Deduction" section?

Why are you asking about that?   Did you already e-file and say you could not be claimed?   Is your parent(s)'s return begin rejected now?    If so, they have to file by mail, and you have to amend your return to change the way you answered the questions in MY INFO regarding whether you can be claimed as someone else's dependent.

The person whose e-file is being rejected as a result of the duplicate use of your SSN will not be able to e-file this year; waiting for you to amend will not help them.   You should amend, but they should mail their own return that claims you as soon as they have it ready to put in an envelope.