I'm being fined for not having health insurance but I was on my parents plan before I got health insurance once I started my job. I was in undergrad up until June 2016

Your parent's plan coverage for you counts toward the insurance requirement.

If you had coverage for the entire year (on your parent's plan, then your own) with no lapses, then answer 'yes' on the page, Did you have health insurance coverage in 2016?

Answer no, if there was a gap in coverage at any point in the year. Then proceed to enter the months covered.