The screen replacement option is only for some models of iPhone, is only available in some stores (and not all countries) and is a quite new option.
I have never known Apple to offer repair of a battery for any iOS device. Apple's battery replacement service for iOS devices involves replacing your entire device with a factory refurbished unit (as clearly explained in their support section on the iPhone battery replacement service). They do not service the battery in your existing device.
Other than that, no, they do not repair, they replace with service units, and send the exchanged item back to the manufacturer for recycle or refurbishment. Refrubished units all have new screens and new batteries.
And regardless, the initial answer to the original post is that there is no legitmate source of new original equipment parts for any iOS device. Even for those devices Apple is beginning to initiate in-store repairs for (the iPhone 5s and 5c were in part specifically redesigned to accomodate in-store repairs of some things), they will not sell just parts to anyone.