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I applied green card for my mom , and my dad does not want greencard
question: 1 only need birth certicifcation for mom, no need parent marriage certification?-Correct.
2 mom enter usa as B2/1 visa, but she stayed here couple years, what we should do about that?
-Nothing special. Her illegal stay will not affect her I-485 approval.

just pay some punish fund? -No. Do nothing.
thanks