Microchip, PIC, Pickit, and MPLAB have nothing to do with Atmel or AVR. Not compatible; before 2016.
Since 2016, there has been integration into MPLAB and a move to an AVR architecture similar to PIC, but of course the ATtiny2313 is an Atmel-era design.

I don't think MPLAB supports programming with Pickit3.
See attached file about using a PICkit 2 to program AVRs using AVRDUDE. If you've got a PICkit 3, I believe AVRDUDE needs it to be put in PICkit 2 mode using PICkit3 Programmer Application and Scripting Tool v3.10 and you will need to revert it to PICkit 3 mode aka: MPLAB mode afterwards.
I don't know for sure if the same connections will work natively with a PICkit 3 under recent MPLAB X, but if the device support matrix shows PICkit 3 support for your AVR, its a fair bet they will, and its unlikely to damage anything so worth a try.
nice information, might be able to get by with AVRDUDE.