My parents are very ....... married.
I've seen "happily" is used in sentences like this. but I want to know that is it correct if we use "happy" instead of "happily"? Why?
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You should use "happily". This adverb modifies the adjective "married". Adjectives modify nouns; adverbs modify verbs, adjectives and other structures. So you need an adverb here.
There is an alternative in which "married" is a complementary participle phrase to the sentence "My parent's are very happy". For example you could say "My parents are very happy living in France". That example uses a present participle, but in theory, a past (ie passive) participle could be used too. But this isn't the sentence you want to say, and in practice we don't use "married" like that. So feel free to ignore this paragraph.
Your example would be
My parents are very happily married.
Because that is the common form of that phrase and happily is the adverb form.
My parents are very happy married.
works as a short form of something like
My parents are very happy [now that they're] married [after dating for 10 years].
In this case, it would be spoken with a distinct emphasis on married.