Why do you say "obojga" here? Which case is it in?
Szukać (szukam, szukasz, szuka) takes the genitive case, and here obojga is the genitive form for oboje.
What's the difference between "oboje" and "oba"?
See my comment here: https://forum.duolingo.com/comment/19529552$comment_id%3D19554885
I read your very helpful comment. Thank you. Is there any good system for remembering all the cases? Or is it just something you have to repeat and learn by heart, like an irregular noun or verb?
If i am looking for something else, what it is a neuter noun and plural, the sentence also starts with "Szukam obojga moich...) Isn´t it?
who would say this? i think parents are always both and not a single parent?
If you need to emphasize the fact that you are looking for both of them:
-Are you looking for your dad?
-Actually, I'm looking for both my parents.
In this case it would be wrong to just say: "No, I'm looking for my parents."
"both my parents" as an object case is a little weird. Standard spoken English should be "both of my parents".
I think either is fine, but "both of" sounds more natural to me.
Why can't i just say "i'm looking for my parents" ?
You can, but the emphasis would be different. This sentence specifies "both" to add emphasis to the fact that you're not looking for just one of them.