No, normally you cannot remain on your parent's policy once you move out of the house. No matter your age (whether you are 17, 25 or 45) if you live with your parents and are a licensed household member than you can remain on your parent's insurance policy usually. However once you move out of the household than you will need to obtain your own insurance coverage since your parent's insurance company will not typically allow you to remain on their policy.
If you own a car than you need to obtain your own car insurance policy for the vehicle at your new address. If you do not have a car and at your new address there is not a car for you to be on the insurance policy of than you should be able to get a non-owner's policy. If you are driving a car titled to your parents you will need to see if you can get your own policy on it even though you do not have insurable interest on it. Some insurance companies such as Esurance do allow this while many do not.
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When moving out from your parents and going to a different address, and especially now that you are going to live in a different city, your rating and risk factors differ so likely when you get a policy of your own it will cost a bit more than the rate you were being charged to be on your parent's policy.
If your parents tried to keep you with whatever car you drive on their policy without informing their insurance company about you moving out of the house this may cause issues if there is a claim since neither you or the vehicle were staying at the address the insurance company thought you were. Also not notifying the insurer of changes like this could be considered misrepresentation to the insurer, a form of insurance fraud.
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