My friend is 16 years old and his girlfriend is 20 years old? Can the girl get in trouble for having sex with the 16 year old? And do they have to have their parents approvement? What is the age different law in the state of North Carolina?

In North Carolina, a person can no longer be a victim of statutory rape at 16 years of age; however, at 16 you are not legally an adult and you are still under the care and control of your parents. If your parents object to the relationship, they are within their rights to take legal action to prevent it. Some do, some don't. This would require a judgement call on your part. As a minor, your parents have legal responsibilty for your safety, welfare and control.

[1]The Constitution gives "heightened protection to a parent's right to direct a child's upbringing." "As the Ninth Circuit stated in Calabretta v Floyd (1999)" "the government interests in the welfare of children embraces not only protecting children from physical abuse, but also protecting children's interest in the privacy and dignity of their homes and in the lawfully exercised authority of their parents."
[1]Source: Findings: Parental Rights by Brooke Everley.