Is it possible for people under 18 to have plastic surgery/cosmetic surgery without parent consent?

Sometimes: It will take a court order. Hilary Clinton as a young lawyer wrote a brief defending a court's right to require religious-nut parents to have their teen's cleft palate repaired. She was criticized for this. The teen may need to engage an attorney in any case, but a sensible physician may be able to bring the folks around.

No: And unless it's something like a cleft palate that should be held until over the age of 18.