Depends on the mechanic.  If he'll stand behind the parts and the work, then be prepared to pay retail plus his profit.  I've never run across an honest mechanic who didn't welcome an owner bringing his own parts.  But they have all made it clear that they, personally, do not guarantee parts that they do not provide.  Get a warranty statement in writing.  And remember I said limited lifetime warranty, not just limited.  Buy them locally so they are easily returnable if need be.  

100% let the shop do it. That way the labour is warrantied as well.

It depends on how much you trust the shop and how honest they are. The mechanic I use I trust him enough to ask for the old part but he still keeps it to show me and if I want it back. But in any other case I would say tell them to keep the old part and that you want it. If they resist it that's not a good sign.

Most places have a place on the work order to check off to keep old parts, yes or no.

If you bring your own parts and they turn out to be defective you're on the hook for the additional labor to R&R the defective new part(s). If the mechanic provides the parts and there's a problem he'll have to do the replacement on his own dime.

It's very common for aftermarket and even some OEM parts to be defective right out of the box. How lucky do you feel?

If you buy parts with a limited lifetime warranty, yes.

It really depends. 

I’ve taken stuff to my mechanic, and he takes care of everything.  With a warranty. 

I’ve taken stuff to my mechanic, and I supply parts, but he won’t warranty the work or parts. 

It all depends on how comfortable you are and how comfortable the mechanic is. And how much risk you and/or the mechanic is willing to bear. 

Really they can't say no. If they do, take your business elsewhere, that's not a good sign. The parts came out of your car, they're still your parts. Unless it's something that requires a return for a core charge, like a torque converter or valve body, power steering pump or something like that, they should give them back if you request. When I had my transmission rebuilt a few years ago I asked for the parts back. They gave them to me in a little cardboard box. Gaskets, solenoids and stuff like that I threw out, but I kept the friction discs, steels and the overdrive planetary gearset. They said they would have given me the torque converter and valve body too, but they had to return those for the core fee.

Posted by: @frankwhite555

does the shop always give you back the old parts?

No, unless you ask.

And often it will cost you more money to keep the old parts (core charge). Ask your mechanic to explain it to you.

It depends on the shop, in my experience. My girlfriend's dad's fuel pump went out in his 2000 E-350 a few months ago, the shop wouldn't install anything except an OEM Ford pump and warranty it if he sourced his own parts. He bought the Motorcraft fuel pump from a local Ford dealer and delivered it to the shop. Some mechanics get annoyed when you give them parts, some don't. 

The shop we use welcomes customers to bring their own parts, but they only guarantee the quality of work being done, not the parts, in most cases. 

Quick and simple question. When you take your car to get worked on, does the shop always give you back the old parts? If not should I ask?

Depends. If there is a core charge, no. If there is no core charge, then yes.