A Disneyland green card, also known as a priority card allows the card holder and up to four guests access to attractions via the disabled entrance.
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As such, you can board the cruise with other forms of ID, including a passport card, a trusted traveler card, a REAL ID, etc. Minors can also travel without passport books as long as they have their birth/ naturalization/ citizenship certificates.
Generally speaking, adjustment applications will need to obtain an advance parole document in order to travel outside the U.S. while their Form I-485 (the application for a Green Card, or adjustment of status) is pending.