See clip from Attorney Rajiv S. Khanna's conference call video that addresses this question.
This is one thing very common in many cases. The issue of applying for a green card, when you are visiting on a tourist visa or even converting a tourist visa to H-1, L-1 and student visa is always a problem.
When you try to do that the government can bring the issue that when you entered USA on a tourist visa it was your intention to convert to a long term status which is illegal according to the government. There is actually a 30 - 60 day rule which says if you enter on a tourist visa and you changed to a longer term visa and status within 30 days of entry, will presume it that you had a preconceived intention to do so. That can be considered to be a fraud and if you’re not able to rebut that fraud presumption that can become a permanent bar from entering the USA.
So if you try to change statuses in 30 days of entry and if you do it within 60 days they can still question it but there is no automatic presumption. If you do after 60 days they may not question. But remember even if you try to change status after 60 days you can be questioned by USCIS, CBP or even consulates about your intention. So when your parents are visiting it is important that they don’t have an intention to apply for a green card when they enter on a tourist visa. If after staying here for 3-5 months it so happens that their intention changes I don’t see any problem with that. Remember you the child, you the son or the daughter what your intentions is I do not care but your parents should not have the intention to convert into green card. If they arrived with the preconceived intention to change to green card they can have a problem. So don’t let them have that intention. After a few months if they are living here and you are able to convince them to file their green card then I think that’s ok.
As for the question regarding birth certificate it should not be an issue because having the maiden name is common in USA as well as India. That should be ok even though her passport contains her married name and her birth certificate contains her maiden name. Under Indian laws - under Hindu Marriage Act you are not required to have any marriage certificates when you are going through a religious marriage. It only has to comply with the laws of that religion. Having a marriage certificate under religious laws is not required.
Regarding marriage certification you can get affidavits from two people who attended the marriage.
As for the question about your father not having a birth certificate you do need to get a certificate of non availability from the place he was born. There is no particular format to the certificate of non availability. All it has to say is that they have looked at the records. When you have that you can get affidavits of two people who were alive when you father was born. So certificate of non availability combined with two affidavits will be required.