Learn Italian: I miei genitori possono restare in questo albergo.

Restare (to stay) is an infinitive (the basic form of a verb). When you use a modal verb like potere (can / to be able to) to say what someone can do, only the first verb changes to match the person. Here, possono matches "they can" (the parents), and the second verb restare stays in the infinitive.

Does restare mean "to rest"?

No, restare is a false friend! It looks like "to rest", but it actually means "to stay" or "to remain". If you want to say "to rest" in Italian, you would use the verb riposare.

I thought family members don't use the article il or i with possessives. Why is it i miei genitori instead of just miei genitori?

You are right that singular, close family members drop the article (for example, mio padre for "my father"). However, when talking about family members in the plural, the rule changes and the article comes back! So we must say i miei genitori (my parents).

Could I say quest'albergo instead of questo albergo?

Yes, you will often see both! Questo can drop its final -o and add an apostrophe before a noun starting with a vowel (quest'albergo) to make it flow faster when speaking. However, keeping the full word (questo albergo) is also completely correct and very common.