In general, individuals granted U.S. permanent residency are eligible to apply after five years of U.S. permanent residency assuming they meet certain residence, physical presence, and good moral character requirements. The law also allows individuals to apply up to 90 days in advance of their fifth-year anniversary. Like most aspects of U.S. immigration law, there are exceptions and certain restrictions, so be sure to consult with an immigration attorney prior to filing.
I do not recommend that one file an H-1B petition without the assistance of competent immigration legal counsel. For one, the petition is employer-based, so it is the U.S. employer that must file the H-1B petition. While the employee is the beneficiary, the employee does not typically sign any of the paperwork. As in many types of U.S. immigraiton matters, the process is a literal legal minefield and the federal government routinely moves the mines. Moreover, the implications of a misstep...
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In order to answer your question, I would need more information, but if you entered without authorization, your parents are unauthorized, and you have no other family (e.g. US citizen spouse) in the United States, your options may be very limited. However, immigration cases are very fact specific. A slight twist of the facts and your situation suddently becomes much different. For example, if you have a grandparent who was born in and lived in the United States, that might change things....