Duolingo, where did you learn your English? ‘Peter sings better than they’ should be accepted.
Yes, 'Peter sings better than they' sounds wrong to me. As far as I know 'them' is used when drawing comparisons (same with 'her' and 'him'. English has some oblique cases too!). "Peter sings better than they do" is grammatically correct, but to be really pedantic the structure is different than the given sentence (it has two verbs) so I think the given translation is the best.
'Peter sings better than they' is also correct... Maybe a bit similar to answering, 'This is she,' if someone asks, 'May I speak to (Mary)?'
Interesting discussion on the subject. https://www.grammar-monster.com/lessons/than_I_me_than_he_him.htm