Check with your state's Dept of Insurance.  You will likely be eligible for some form of assistence. 

If you want to purchase a commercial policy, you should consider a comprehensive medical plan with a HIGH deductable ( $1200+) to cover catastrophic events, and a tax qualified Health Savings Account for routine care. 

Your parents are right. It's the responsible thing to do to budget for your medical care.  However, because young adults often have limited income, it is not uncommon for parents to help with the premiums.