Hopefully you'll land a job that includes health insurance. But if not - or if your coverage won't kick in for a period of time - you'll want to consider other health insurance options.
The first question to ask yourself is "Are you in good health?" If so, you can likely qualify for a short-term medical plan. This will cover you for a specified period of time, and is generally the most cost effective way to "fill the gap" while waiting for your other insurance to kick in.
Typically, short-term policies are written for two to six months and are renewable only once. They cover hospitalization, emergency and intensive care, as well as necessary expenses for related services performed outside the hospital -- like X-rays and lab tests.
But if your health is not so good, there's a chance you might not be accepted by the health insurance company. In this case, you will want to exercise your COBRA option on your parents' health plan. In short, if you are eligible for COBRA coverage, you are guaranteed coverage under your parent's plan for up to 18 months.
The problem with COBRA coverage, though, is that's it's generally more expensive than other insurance policies. So you might want to find out first if you can qualify for a short-term medical plan and go with COBRA as the last resort.