If you are going to be put on your parent's auto insurance policy as a driver then your driving record will be a rating factor and thus could affect the rates.
As a licensed household member and listed driver on the car insurance policy the insurance company will want to check your motor vehicle record (MVR). Any tickets you are convicted of and accidents you may be involved in will be placed on your driving record which the insurance company would see as part of your MVR.
Depending upon the insurance company's rating system the tickets or accidents could affect your parent's insurance rates. There is not a way to be on the policy but not have your driving record used as part of the rating process since you are a driver of the cars listed on the policy.
If your parent's insurance rates were to rise due to your driving record they may require you to get your own policy or they could shop around for auto insurance rates with other car insurance carriers.