It will depend on many factors.
First, you might be able to let the junkyard take your car and still let you take some things. My ex's Tercel got totaled by hail damage. The buyback was $200. No-brainer. I accepted the $3500 (minus the $500 deductible and the $200 buyback). They gave me a check for $2800 and she drove a golf ball for another three years. They basically bought the car but I kept the [salvage] title.
That car later got actually totaled. A trucker lost a wheel which she hit and ripped off the front right suspension which sent the car into a concrete wall. She was fine, but the car was wasted. I showed up while they were clearing the accident and gave the tow driver the title, and they had no problem with me stopping by later and taking the good head unit out of it.
They are accepting the car to most likely be a crush credit. They don't care if it has factory steelies on it or 20" chrome rims. Some 'yards might want to hang onto the goodies to sell for a little profit.
- what is the buyback price? Most insurance companies assign their own values. Once you find out that price, just do the math. Will the money you get selling the goodies be more than the buyback?
- check with your insurance company. Did you claim the aftermarket upgrades as part of the policy? If not, they are insuring a stock vehicle and they don't even know about the upgrades. How that plays out (whether or not you are allowed to keep them) is up to whoever is towing the vehicle, unless the insurance agency steps in to claim insured property.
- Do you want the headache of parting out a car? If the buyback is $500 and you think you can get $1000 for the parts you sell, is it worth the labor of getting those parts off, and the disposal of the vehicle when you're done? That's all up to you.
- what are the insurance laws in your state. In CA, the car was mine until I sign and hand over the title, so theoretically I could have had it towed, but kept the title so all of the things in or on the car still belonged to me. In my case, they didn't care. I could have removed the engine if I wanted. They were just going to crush it for credit. In PA, the car ceases to be mine when I accept the claim (or cash the check after the claim... I forget). At that point, the car belongs to the insurance company regardless of who holds the title. Once the insurance company pays me for the car, it is theirs. The title is immaterial because the junkyard can get a salvage certificate for $14 regardless of whether or not I give them a title.
- is the claim already done and accepted? In many situations you can deny the claim if you've yet to accept it. You can initiate the claim, hear the numbers, and say "no thanks, I don't want to report this claim." Then nothing happens. You still own the vehicle without a salvage title and you can do with it what you want. Take the wheels and pretty things and then file the claim. Due research needs to be done though before you play with insurance laws and the terms of your contract with your insurance company. Don't be afraid to ask. Many times insurance claims are handled by a third party. When they have a totaled car, they "sell" to a sub-contracted company to handle the details.
99% of the time, insurance companies are dealing with stock vehicles. It's a slam dunk. Here's your check, car gets towed to a junkyard, go buy another hoopty. When they get a "special" situation like yours or mine was, they might completely balk at you taking parts, or they might just say "take what you want." Best practice I think is to just ask before each step.