In almost all states (definitely Arizona, Connecticut, Florida) you have the right to contract with the body shop of your choice - the insurance company cannot require you to go to a specific shop.

You have the right to NOT fix your car - you can pocket the money and just drive it or junk it. However, if you choose this option, your insurance company might not fix it again if there is a subsequent claim.

You also have the right to have your car fixed properly - and while that might allow for LKQ (like in kind and quality) parts to keep some costs down, it shouldn't be for "all" recycled parts.

I handle several insurance claims per month (I own a motorcycle repair shop). In most cases, the adjuster writes up all new parts, and then if I want to make a deal with the customer, that is up to us.

I agree with Steve - well run shops don't let customers supply parts (except under special, rare situations). But, a well run shop should work with the customer to get what they deserve, and as close to what they want as possible.

I also agree with Steve that there is something funny here...

-David