I am a teacher of English for the British Institute in Chile and, yes, "at the weekends" is what we teach. Duolingo should accept it.
'I have no time at weekends' is a more commonly used way of saying this in English; is it an accepted translation?
Not a native here, but been living between the UK and Ireland for 20 odd years now, and I would definitely say "at the weekend/s". It is possible I may be wrong, but my instinct is that it is right.
I am a native English speaker who has lived in England most of her life. The English would never say "on the weekends". "At the weekends" is totally correct.