My parents just immigrated to the US and stayed with us. Also, my parents has green cards . They are old and not working now. They stayed in Texas for 6 months and went back to Vietnam for visiting for another 6 months. Can I still claim them as dependents and if yes, how can I file the claim?

Thanks

Yes, you can claim your parents if then did not earn more than $3,500 (for each) in 2008 and you provided more than half of their support for 2008.
Read about Qualifying Relative requirements: Your U.S. Tax Return: Requirements for claiming a dependent

Agreed. Their green card status makes them U.S. residents for tax purposes.

My mom 89 is years old and has lost her passport, and green card. She has no way to show proof of who she is to replace these items. What can she do to replace these items. She entered the us over fifty years ago?
Thank you

The U.S. Customs and Immigration Service (USCIS) has a computer file for your mother. She may need to go to a local USCIS office to verify her identity, but they can replace the green card.

I suggest you call the local USCIS office, explain the problem and get advice on how to proceed.