According to the New Jersey Motor Vehicle Commission, you may use your foreign country's driver license as proof of driving experience if you have an International Driver License issued by your country or if your country is a member of the United Nations Convention on Road Traffic (which Mexico is) and your license is translated into English.
A valid Mexican driver's license may allow you to drive within the United States (though you would need to check with each state's DMV since laws can differ) but typically insurance companies within the US will require you to obtain a valid license from a State within a short period of time of starting up an auto insurance policy.
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Your parent's would need to check with their insurance company to find out if you could be placed under their policy. Likely even if you are put under your parent's policy as a driver, you would need to still acquire a US license. If you are going to reside in New Jersey, the NJ MVC gives information on how turn in a foreign license and apply for a NJ driver's license.
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