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doesn't matter if its a guy or girl or age just something in that genres
I'm sorry, I love you
Tree of Heaven 
Both dramas are sad though
Gomen, Aishiteru
not kdrama, not jdrama, but if you have netflix, you should watch "first they killed my father"
There's actually quite a bit of drama with characters searching for their parents, but most of the time it's a background story. 
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. 
In Lucky Romance the second male lead return to Korea because he wants to find his father
Dramas that actively is looking for their parent (parentage) from the start:
Birth of a rich man
oh my geumbi
my Sassy Girl