First ...you do not want to plead guilty to petty theft. A petty theft conviction will have long lasting effects on your future including disqualification from future employment. That being said, your best chance of avoiding a conviction is to hire an experience criminal defense lawyer.
As for eligibility for the public defender, this is determined by your financial situation. If you do not or cannot afford to hire a criminal defense lawyer, then you need to go to court and request to have the public defender appointed to represent you. The court will then have you fill out a financial application which will determine whether you qualify or not. From what you describe, it sounds like you may.
My suggestion is to do whatever you can to hire a criminal defense lawyer. While I understand that you parents are unwilling to help you here, it would be worth your while to ask for their assistance in this situation given the potential consequences to your future should you be convicted. Maybe they would be willing to give you a loan for the attorney and let you repay them over time. The bottom line ...do whatever you can to make it happen.
I hope this answer was helpful.