Can I petition my parents if I just became a US citizen last month?

In general, after an adult becomes a naturalized U.S. citizen, the citizen may apply for his/her parents to adjust status to become Lawful Permanent Residents (to get "Green Cards") if the parents entered the U.S. lawfully and with inspection. This is true regardless of whether the parents may have overstayed their visas. Also, immediately upon filing an adjustment of status application, the beneficiary's new status becomes "applicant for permanent residency," which authorizes the beneficiary to remain in the U.S. while the case is pending, regardless of the expiration date of the previous visa. Note, however, that where the parents entered the U.S. with visitor visas (or certain other temporary/nonimmigrant visas), an application filed very soon after entry may create significant problems. Visitor visas are for people who genuinely intend to enter the U.S. to stay temporarily in compliance with their visas and then depart. When an adjustment of status application is filed very soon after entry, it may appear that they entered with the actual intention of applying to adjust status and remain permanently. There then becomes a risk that the USIC could view the use of a visitor's visa as constituting visa fraud, with very harsh consequences. It would be wise to engage an immigration attorney who, upon learning all of the relevant facts about your parents and their immigration-related history, will be able to advise about immigration eligibilities, options and strategies. Some immigration law firms, including mine, offer legal services on a "flat fee" basis so that a client will know the total expense from the very beginning, and a few immigration law firms, including mine, offer an initial consultation free of charge.