Do you need to tell your parents? No. Should you tell your parents? Yes. You very well may be an adult, but having the support of parents and loved ones certainly will help through what can be a difficult time. Being accused of plagiarism at college is incredibly serious. You should not go through the process alone. You cannot depend on the school to do the right thing, be it the professor, the conduct office.
If you're accused, you shouldn't engage with the school in any capacity without taking the necessary precautions. Every step of an academic misconduct case is critical at the early stages, especially because they could set the stage for later aspects of the case. You should tell your parents. You should also have an experienced academic misconduct attorney advisor in your corner from as early as possible in the process because they can help you best understand and navigate the process.