Learn Italian: Che cosa cercano i tuoi genitori? - What are your parents looking for?

Che cosa literally translates to "what thing" and is the standard, most grammatically complete way to ask "what" in Italian. In everyday spoken Italian, people often shorten it to just Cosa or Che, so asking Cosa cercano i tuoi genitori? would also be perfectly correct.

Where is the word for "for" in this sentence?

The Italian verb cercare means "to look for" all by itself. You never need to add a separate preposition for "for" after it. So, the single word cercano completely covers "they are looking for".

Why is there an i (the) before tuoi genitori (your parents)?

In Italian, you almost always need a definite article like i before a possessive word like tuoi. There is a special rule where singular family members drop the article (for example, tuo padre), but since genitori is plural, it requires the article.

Why is the subject (i tuoi genitori) at the very end of the question?

In Italian questions that start with a question word like Che cosa, it is very common and natural to flip the English word order and put the subject after the verb. Placing it at the end sounds much more natural to an Italian ear than putting it before the verb.