What is 'Class of Admission' on the form I-131 (Advanced Parole)?

"Class of admission" refers to the visa category under which the person applying was last admitted to the US, which in your wife's case appears to be H4.

Thanks for your inquiry. I would follow the advice of my colleagues below. The concern here is that the Form I-131 is used for a number of different reasons. It is the application filed for advanced parole, a re-entry permit and a refugee travel document. So when the form is used for more than one purpose, some of the fields which appear on the form are not necessarily applicable. And when you go through the I-131 you see that there are specific portions of the form which only apply to Advanced Parole applications, portions which apply to refugee travel documents and portions which apply to re-entry permits. Would be a heck of a lot easier if CIS would make the applications consistent with the purpose for which the form is filed, but it is CIS. "H-4" or "I-485 Pending" or both is fine. As long as you complete the form correctly, indicating that this is an application for advanced parole, you should be just fine. And if you have questions or you are concerned, provide an explanation sheet with the form (can add it as an addendum). In your case, it should be abundantly clear that you are applying for advanced parole because the other travel documents either require that you were granted resident status or granted asylum. So absent documentation to establish such status having been granted, the adjudicating officer should be quite able to figure it out. Good luck.