Translation:Your parents care about you.
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In this sentence, 很 is not a meaningless filler word. It is an adverb that intensifies emotional or psychological verbs.
Noun + 很 + adjective = 我很高兴 = I am happy
很 + emotional verb = 我很喜欢 = I really like
Since 关心 is a verb, 很 (an adverb) translates to really, very, a lot, etc.
Certain "psychological verbs" related to feelings or emotional sates can be modified with 很 (hěn) to increase their intensity. This is similar to using "really" or "very much" in English. Remember that this only applies to psychological verbs, such as: