A determination of a person qualifies for a public defender the court will look at the income of the individual. I does not matter that your parents claim you on their taxes. The court will look at your income and your ability to afford private counsel.
In order to have an attorney appointed, you will need to court and ask for one. This usually occurs at your first court appearance, called an arraignment. The judge will ask you if you want a court appointed attorney, will ask you questions about your finances, and may have you fill out a form regarding your income and expenses. It will be helpful if you show up to court prepared with this information.