Local Tax Confusion

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I am a CPA, licensed in PA. In order to answer your question, I need to know what state(s) you and you parents live in. But, generally, you consider your parents home as yours for local tax purposes and thus pay taxes there. In PA however the local/school tax is only on earned income so poker winnings would not be taxable unless you filed as a professional, which given your situation you probably will not.

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my parents home is in Pickerinton, Ohio. I have been going to college/living in Athens, Oh. I am just talking about the city tax there, and the school district tax. the tax lady I have talked to said I should just file as being self employed

Filing as self-employed subjects you to self-employment tax, basically 15% of business earnings on top of your income tax. Unless you have substantial losses or other expenses, it is probably not worth filing as SE'd. According to the directions on Pickerington's tax form they specifically state that "Prizes and gifts not related to employement" are not taxable. I would say that your tournament winnings fit that definition and are thus not taxable.
Also Pickerington's tax rate is 1% while Athen's rate is 1.65%, but I guess it is the school district tax in Pinkerington (1%) that yields a higher rate. Athens specifically includes gambling income in income, so if you file in Pinkerington and only pay the schoold district tax of 1%, you will still be ahead of the game.

As I am not licensed in OH, you still should talk to a tax adivsor in your state.