looking for kdrama or jdrama about guy or girl searching for parents

I'm sorry, I love you
Tree of Heaven 
Both dramas are sad though

not kdrama, not jdrama, but if you have netflix, you should watch "first they killed my father"

You're Beautiful was the first one that came to mind. The girl kept searching for her birth mother.
The Legend of the Blue Sea is similar. The guy is looking for his missing mother. Antonin is right, though. Those themes are not central to the dramas I mentioned. 

Dramas that actively is looking for their parent (parentage) from the start:
Birth of a rich man
oh my geumbi
my Sassy Girl