It's Genitive. Szukam kogo? czego? (I search for whom? what?)
But as for the 'why', I don't know if there is any good explanation. Obviously, we Poles just know that it's Genitive (and if not, we check what questions does the verb answer, as above), just as you know what English grammatical construction to use. So we remember which verb takes which case (sometimes different ones depending on the context). I guess foreigners, at least non-Slavs, just need to remember...
But maybe there's a possible help in remembering if you think of it as a "Search of parents", even if that's not grammatically correct?
Nowhere. In Polish you use a lot less possessives than in English. Especially with family members. You just assume that the family member belongs to the subject of the sentence. And if it's not the case, then you specify it (Ja szukam twoich rodziców = I am looking for your parents).