You will in complete the Health Insurance section of Turbo Tax. If you both were covered all year, you will indicate that "We all had health insurance coverage all year."
If your parent's health insurance was private/employer insurance, there is no additional information you need to enter for this insurance coverage.
If the insurance you got after graduation is through your employer, there is no additional information you need to enter for this insurance coverage.
If you got Obamacare/Marketplace insurance after graduation, you will indicate yes to " Was anyone enrolled in any of these less common plans in 2016?" You will then enter the information from your 1095 A form. This information is only entered for the months you were covered by Obamacare/Marketplace insurance.