By part exchange do you mean trade in?
I always sell private party. It's more work but you get way more money. For example, my girlfriend didn't want to deal with it so we just traded in her old 2014 Kia soul on a 2020 Jeep renegade. The Kia had 100k miles on it and they gave us 3k. In my area they are going for $5-7k.
Another example with my old 2006 F-250, I wanted a newer truck so I went to the dealer and they offered me I think it was $7k as is. I took everything back to stock (wheels and tires, lights, bumper, etc.), sold it for $13k and sold everything else for another $2-3k.
Also as far as buying goes, I also try to buy private party for 2 reasons. Number 1, they know the vehicle and might have service records. You can kind of pick up on if they are a dodgy person dumping a crap vehicle or actually selling a decent vehicle. The other reason is price. My old 2011 F-350 I bought for $20k private party. From a dealer that would've neared $27k.
There are a couple down sides to private party sale, the most obvious is once it sells you don't have a car and you are kind of rushed shopping for 1. The other is the payment aspect. You have to hope that they are giving you a valid check / real money. I always have a currency pen if I get paid in cash, and if they pay me in a check I make sure it's a cashiers check and I like to be present when they get it then immediately put it in my bank account before signing the title over. The other thing to think about is if they wreck on the test drive. It's your insurance that covers it so you have to pay the deductible and you have a crash on your driving record even though it was their fault and they were the one actually driving.
All in all, I prefer to do everything private party because I pinch every penny possible. There were 2 times I did a dealer trade in, 1 was because it was the perfect truck (my 2006 F-250), and the other was my girlfriend didn't want to deal with the hassle and didn't have the time as she works in health care.