No, it is not likely that your parent’s will be able to continue to have you and your car on their insurance policy if they move out of state. The state of Illinois has different insurance liability requirements than Wisconsin is one problem that may arise but also most all insurance carriers will not cover a person and car that is not part of the household. Since your car will not be garaged at the address of your parents then an insurance company will not usually permit you or your car to be covered on your parent's policy once they move, especially to another state.
Illinois state law requires minimum Bodily Injury Liability coverage of $20,000 per injured person up to a total of $40,000 per accident, and Property Damage Liability coverage with a minimum limit of $15,000. This basic coverage is often referred to as 20/40/15 coverage. Illinois state law requires a minimum UnInsured / UnderInsured Motorist coverage of $20,000 per person, $40,000 per accident for any bodily injury caused by the uninsured driver.
The state of Wisconsin has a financial responsibility (FR) law that applies to any motorist licensed to driven in WI. The easiest and most often used method of meeting the FR mandates is for a motorist to purchase car insurance for their vehicle.
If a driver or vehicle owner decides to comply with the FR law by choosing auto insurance there are minimum liability limits set by the state of 25/50/10. These numbers stand for $25,000 for one person in one accident injury or death; $50,000 for injury or death of 2 or more people in one incident; and $10,000 for property damage caused in one accident. The Wisconsin insurance also requires uninsured motorist bodily injury coverage of $25,000/$50,000.
So as you can see if your parents only obtained the minimum IL auto insurance required by that state it is not enough to comply with the WI financial responsibility law auto insurance limits.
If your parents move to Illinois and you remain in Wisconsin then you will need to obtain your own car insurance policy for your motor vehicle. Since you are a young driver rates are more expensive, due to your inexperience as a driver from the perspective of insurance providers.
You can see if there are ways in which to reduce the premium you will pay. You can look for discounts due to safety features on your car for one. Other ways in which to get discounts may be taking a driver improvement or safety course, some insurers lower rates if you take approved traffic schools. Also if you are going to school and maintaining good grades some insurers give good student discounts.