Yes, "Die Eltern mögen Käse" is a correct German sentence, but it is different from "Die Eltern mögen den Käse," hence only one is a correct translation.
The former sentence expresses that the parents like cheese in general, in the latter the speaker might be referring to a specific chunk or perhaps an ensemble of cheese, saying that the parents like this particular cheese that they are currently eating.
The meaning is really the same between the respective German and English translations of the two sentences.
"Den" is a declination of "der", which is the masculine definite article, that goes with Käse, which is also masculine. More specifically, "den" is the accusative masculine singular declination of "der".
The verb "mögen" requires all items that one likes to be in accusative. Coming from English learning these declinations can be daunting, as they are practically invisible in English, even though they are also there.
Take, for instance the sentence "I like the dog". "The dog" is in accusative, but you can't tell (other than by context), because the definite article looks no different than if you said "the dog likes me." Yet, in the latter sentence "the dog" is the subject of the sentence, hence it is in nominative case.
In German these two sentences translate to "ich mag den Hund" and "der Hund mag mich." So, you can tell the different case "Hund" appears in, just by looking at the articles "den" (accusative) and "der" (nominative).
If we use a personal pronoun in the English sentences, for instance, "he", we can see the different cases too:
He (the dog) likes me.
I like him (the dog).
Notice the two different forms "he" (nominative) and "him" (accusative).
Er mag mich.
Ich mag ihn.