"They don't allow their children to play outside the house in the evening."
Translation:Ili ne permesas, ke iliaj infanoj ludu ekster la domo vespere.
It could never be ludi (infinitive), because the verb has a subject, 'iliaj infanoj': (They don't allow that) their children play... But why it is ludu (imperative) instead of ludas (present) or ludus (conditional), I can't tell you. In Spanish they would say 'No permiten que los niños jueguen...' - a subjunctive, that is. But this counterintuitive imperative in these sort of sentences is one of the abiding mysteries of Esperanto...
Why is "Ili ne permesas siajn infanojn ludi ekster la domo vespere" wrong" (because of "siajn infanojn" instead of "al siaj infanoj")
Because "permesi" does not take the person as a direct object. If it does take a direct object, that is a subordinate clause or an action.
You can have "permesi iun agon" (to permit a specific act), "permesi, ke ..." (to permit that ...) and/or "permesi al iu ..." (to permit someone to ...)
But verbs like helpi also dont take the direct object but you can say: ''Li helpas vin'' instead of ''Li helpas al vi'' so why ''Ili ne permesas siaj infanoj'' wouldnt work here?
helpi does take a direct object (the person helped) in Esperanto.
permesi does not.
The situation in other languages may be different.
Oni bezonas komon: "Ili ne permesas, ke...".
Sed Duolingo kutime ne zorgas pri interpunkcio.
Do plej verŝajne vi faris alian eraron - eble, ke vi uzis "...ke ... ludas" anstataŭ "...ke ... ludu".
Why not "siaj infanoj"? Wouldn't "iliaj" imply that it is someone else's children, not their own? Or am I misunderstanding how "sia" works?
"Sia" refers to the subject of the phrase. So it can never be part of the subject itself.
But the subject of "ludu" is "infanoj".
So you can't make a distinction here between "They don't allow their (others') children to play" and "They don't allow their (own) children to play".
You could say something like "Ili ne permesas al siaj infanoj ludi ..." -- here, "infanoj" is not the subject any more but the complement of the preposition "al" in the same phrase as the "ili" who are doing the allowing, so you can use "siaj".
Ili ne permesas al siaj infanoj ludi ekster la domo en la vespero. The translation given refers "to their children", why "to"?
The English sentence does not contain "to their children" -- it says "They don't allow their children to play outside the house in the evening."
The Esperanto sentence does not contain "to their children", either -- it contains only Esperanto words, not English ones.
Are you asking about why it is al siaj infanoj?
Perhaps you can think of it as "give permission" -- the direct object of permesi in Esperanto is an infinitive or a ke clause or a noun: the thing permitted. And the person "to whom" that permission is given is introduced with al.
There were two correct translations offered here, and I still got it wrong. "They don't permit for their children to play outside..." and "They don't permit that their children play outside..."
Ili ne permesas al siaj infanoj ludi ekster la domo vespere. Ili ne permesas, ke iliaj infanoj ludu ekster la domo vespere.