Any idea what would be the correct answer for the "class of admission" for TPS person applying for travel document ?

It's on the first page of Form I -131, on right side Part 1, other information, number 6 (after Country of birth and Country of Citizenship)

What is the right way to confirm this info?

I have been adding TPS as a class of admission but i have seen officer note DA while stamping Paroled

Thank you Nash, that was very helpful; any thoughts about the list of documents and items that needs to submitted with Form I -131?

You definitely need either receipt of your TPS application or approval,
2 photos,
copy of your passport,
cover letter explaining why you are requesting advance parole.
Check or money order, I usually fill their Credit card authorization

Does anyone knows how long does it take to get Advance Parole from Nebraska service center my case is pending over 4 months and haven’t got biometric notice yet.

Mine has been pending since September. I got biometric reused notice in October. I know some people that got within 3 months. It totally depends on your luck.

Jai Nepal

Which service center did you apply from?

Apparently AP takes 4 to 16 months

Raj11, Nepal ko sabai Nebraska ma jancha. My receipt notice is from Nebraska