I was a dependant on parent's marketplace insurance. We're filing separately for the first time. Do I include the 1095-A on my taxes or does it go on parent's taxes only?

Yes, you will include the information on your tax return if your parents are not claiming you as a dependent on their return.  They will have to include the information as well and you will have to allocate the policy (if you or they can qualify for the Advanced Premium Tax Credit).