Yes, Nausicaa believes her parents will help Odysseus. There are several reasons for this.
First and most generally, the Greeks believed in the obligation of the host to the guest. This was called "xenia," and they took it very seriously. Odysseus is a guest.
Second, Odysseus is very wily. Even in their short encounter, he may have shown himself charming.
Third, Athena makes him appear like a god. These people were very status conscious.
Fourth, and perhaps more hypothetical, she's of a marrying age, and he's a godlike male. Perhaps she'd charm Daddy?
Awesome answers! :) There are also some other reasons why Nausicaa is so certain that her parents will help Odysseus.
Her parents, King Alcinous and Queen Arete of Phaeacia, are really the kindest and most honorable people. The whole scene in which they host and entertain Odysseus shows this. They are extremely hospitable, even beyond the normal Greek bounds of xenia. Their love and respect for each other and their extremely happy marriage are emphasized a lot. Nausicaa knows that her parents are just really nice people who would gladly take in a random shipwrecked stranger.
Also, Nausicaa herself is super enamored by Odysseus. She has a crazy crush on this mysterious stranger (who, btw, is naked when he shows up). After this, there are LOTS of hints of marriage from both Nausicaa herself and her parents. King Alcinous pretty directly asks Odysseus if he wants to stay in Phaeacia and marry his daughter. All in all, Phaeacia is just a really lovely place.
Thank you for your answer :)