Translation:They don't allow their children to play outside the house in the evening.
In such sentences (mi deziras, ke ...; vi volas, ke ...; ili ne permesas, ke ...) you use the imperative in a similar way to how some Romance languages would use the subjunctive.
If you wanted to use "ludi", then it would have to be in a situation where "iliaj infanoj" is not a subject - for example, "Ili ne permesas al siaj infanoj ludi ekster la domo vespere".
I'm thinking after reading more about the -u ending in ludu is that rendering this sentence in English as "They don't allow that their children should play outside the house in the evening." would be closer to the Esperanto. Even though it's called imperative, the -u ending is for more than just commands. It also expresses strong wishes and desires.
You're indeed right that the -u ending is not just an imperative.
If you want to go for an exact translation, you could also use the subjunctive in English for this kind of thing: They do not allow that their children play outside the house in the evening. Although not noticeable, play here is a subjunctive in English (which is rather similar to the -u ending in Esperanto). If you were to substitute children with child, you'd get the following sentence: They do not allow that their child play outside the house in the evening. Notice how it should be play rather than plays. (The present subjunctive in English is generally only noticeable in the third person singular, or when using the verb to be.)
In Esperanto, if the second part of the proposition (their children playing outside) is expressed as a complete sentence, then it has to be connected to the first part (they do not permit) with "ke". I always think of this as similar to English:
- They do not permit, that their children should play outside.
I question your use of the word "flow" here, mostly because I'm not sure it's meaningful. Taking my best guess, you are asking why you can't translate the English sentence word for word to something like:
- Ili ne permesis al siaj infanoj ludi ekster la domo vespere.
The answer is, you can. If you're looking for a model sentence, try this one from Zamehof:
- nur la fiereco ne permesis al li pasigi la tutan tagon sur la ferdeko
It should not, because ‘iliaj infanoj’ is the subject of the subordinate clause; since ‘siaj’ would refer back to the subject, it cannot be contained in the subject itself. If you just consider the sentence
“Iliaj infanoj ludu ekster la domo vespere.”,
then it becomes clear that it cannot be ‘siaj infanoj’, as this would be the subject ;).