So so... almost correct. Let's go to check them one by one:
In the first sentence:
My parents made me clean my room before allowing me to go out with my friends.
The translation is almost correct. It sounds very formal and quite artificial:
Mis padres me obligaron a limpiar mi habitación antes de permetirme salir con mis amigos.
To me, the verb obligar is excesive. Sounds like a punishment. If that's the case it's alright, but if it isn't is better to say:
Mis padres me hicieron limpiar mi habitación (cuarto) antes de permitirme salir con mis amigos.
The second verb in italics permitir is correct, but sounds very formal in this case, like a sentence from a textbook of Spanish... Wait!!!. At least in my country, would be more natural to say:
Mis padres me hicieron limpiar mi cuarto antes de dejarme salir con mis amigos.
Syrux is right. The correct tense in this case is
preterito perfecto simple because the action is ended. If you say me hacen or me obligan then you mena that it happens everyday, all the time, not only once.
The second one is ok. But we can notice few things. The first, the tranaslation:
Dijimos a nuestros amigos que llegaríamos tarde para no preocuparles.
Souns well for someone in Spain. In Mexico and maybe other places in America, that word: preocuparles sounds strange. Is leism, I know, but to us it sounds like missing an indirect object. We would say this:
Dijimos a nuestros amigos que llegaríamos tarde para no preocuparlos.
Le dijimos a nuestros amigos que llegaríamos tarde para que no se preocuparan.
In the case of tarde is singular and not in plural is because it is not an adjective as you think, in this case is an adverb of time. So the use is correct in singular, always.
About you last question or doubt,
como que isn't correct simply because is a literal translation of the English
so that. It's more like an idiom or a way to express the same idea that in Spanish is suitable the preposition para.