When does the car dealership do a hard pull on your credit?

If they are paying for the car with a personal check there is no loan, no credit check.  

The only one who needs to know your an authorized driver is the insurance company,  the dealer does not care nor will they check your credit unless you cosign the loan

There is no credit check if the car is paid for at the time of purchase. Insuring the car could be difficult with your credit.

I find it interesting that one of your other questions involves defrauding a credit card company.

And you are so very easy to find now that you've posted this on the open internet.

No credit report is pulled when a buyer pays in full with a check. But if financing, the credit report of the legal buyer is the one pulled.

Credit checks are only done on people who are applying for loans. If your name isn’t on the loan, they have no reason to check your credit. The dealership has absolutely no interest in who’s driving the car. 

You aren't part of the transaction at the dealership.

The insurance company needs to know about you, but the dealer (and lender, if one becomes involved) don't know about you. 

Your parents are buying a car in full with cash.   The dealership isn't going to pull your credit because you aren't borrowing any money.  You also aren't buying the car.

"I am in $10,000 in credit card debt and am worried they won't let me buy the car."   You aren't buying a car.  Your parents are.

"Authorized driver" refers to insurance.   It sounds like your parents are buying a car with the intention of allowing you to drive it.

Make no mistake...this is NOT your car even if your parents told you that it is. 

If you crash said car and paralyze a single mother with three kids, your parents will be sued from here to next Tuesday and the 500k liability insurance won't be enough.

Your parents are better off gifting you money to buy a used car for cash in your own name and getting you on your own insurance policy.