I can't see the scope for issues IF there's time in your shift to do it. If there isn't and you don't want the overtime then give them something back.There have been loads of issues at our depot with part-timers working from home especially when full-timers are doing all the prepping, bundling everything up and then taking everything back to the office whilst the part-timer finishes at home.
It's all just work, the idea that it's not your work or it's somebody else's work is you have to admit pretty weak.
If the issue is that the part-timer isn't working their hours that might not feel fair but ultimately it's not really anything to do with you.