you certainly can, if you are a citizen, send your parents a copy of your naturalization certificate. Remember more importantly they need to show that they do not have an immigrant intent
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No, you do not need an invitation letter. Period. If you would like to increase their chances of getting a visa by explaining the purpose of their trip; intentions to return and so on, you should do that. Explain in a clear language and give it to your parents. They will take it with them to the embassy. Good luck.
No, and an invitation letter is not even remotely enough. Your parents must prove to the satisfaction of the consular officer that they have such strong ties to their country that they will not remain in the US.
J Charles Ferrari Eng & Nishimura 213.622.2255 The statement above is general in nature and does not constitute legal advice, as not all the facts are known. You should retain an attorney to review all the facts specific to your case in order to receive advise specific to your case. The statement above does not create an attorney/client relationship. Answers on Avvo can only be general ones, as specific answers would require knowledge of all the facts. As such, they may or may not apply to the question.
Your invitation letter will have little to do with removing the presumption that they are coming to the US with the intent to immigrate. While consular officials can review documents, the interview is not designed for review of evidence. Evidence is primarily provided in the application, and must establish sufficient ties to the country so as overcome presumed immigrant intent.