Care a lot should be accepted because of hen
more often than not, hen is just a meaningless filler word, without it the sentence doesn't sound native.
If you really want to express 'care a lot', I would use 非常关心.
Although I don't disagree I take issue with really/very not being an acceptable translation.
I agree with beaudannar. It also feels more natural to say "Your parents really care about you" than "your parents care about you".
"很" means "very" here because it precedes a verb. However, "Your parents very care for you." is bad English, so a translation like "Your parents care for you a lot." needs to be used.
Your parents really care about you. is an acceptable translation. It doesn't change the meaning of the sentence and you are inconsistent when you are translating the word 很 or just allowing it to be a filler.
Why is the word “concerned” not accepted
It doesn't adequately cover 关心, at least to me. I would probably use something along the lines of 关注, and only use it specifically, say, for studies: 关心你的学业.
Cannot say, father and mother rather than parents???
很, very much, no?