Should I petition my parents for green card if they are coming to the US for only for 1 month?

Nothing happens in one month and if they are coming for just one month, it is best not to do it while they are in the US since nothing will happen and it may jeopardize their future ability to travel to the US as visitors. If they are really intending to immigrate to the US, you need to figure out whether to do it through adjustment of status (once they are in the US for a long period of time) or through consular processing in their home country.

An alien coming to the United States only as a temporary visitor for only one month and who, then upon entry then files a Form I-130 petition to get green card may unwittingly be conspiring to commit immigration fraud and potentially faces any number of sanctions including fraud and refusal to enter and expedited removal right at the port of entry. If so removed, such alien may be unable to come to the US for 5 years or more. While this kind of fraud may be forgiven to aliens with US citizen family members, you cannot guarantee that CIS will in fact favorably exercise discretion. Before taking further steps, please speak to an experienced immigration attorney.

You are asking detailed questions that would best be answered by a consultation with an Experienced Immigration Attorney. The money spent should be well worth it if it helps your family. Many, like myself, will do these over the telephone to accommodate clients.

You or your parents need to discuss their long term plans and goals before filing for a greencard for them. If they only intend to be here for one month, and don't intend to return in the near future, they should not file for greencards at this time.

It is a bit more complicated than you think. You should consult with an immigration attorney before you file an I-130 and have your parents come to the US.

The easiest process would be to apply when they are in the US. However, the process takes between 6 to 8 months from start to finish. They should remain in the US until they receive a travel permit. This process takes about 2 months. If they are outside the US or leave early, then the process takes about a year. And they will have to deal with the american embassy abroad. Much more complicated situation then filing in the US.