No - if you've moved out of your parents' home and taken your belongings with you, then you'll no longer be covered by their contents insurance. This means that you'll need to take out your own policy if you wish to be covered at your new flat.
It's important to be aware that your policy only covers your belongings in your flat and not the belongings of any flatmates you may live with. Similarly, we're only able to provide cover for your legal liability, and not that of any flatmates that you may live with.
Each flatmate will need to hold their own contents insurance policy to have their belongings and legal liability covered.