Can I become a permanent resident if I am living with my permanent resident parent?

If you are unmarried your dad can file the I-130 now. Otherwise he can do this when he is a USC. It will then take many years until you are able to file for your green card. UNless you have eligibility under 245i, you will not be allowed to adjust in the US because you are not considered an immediate relative under the law. So you will need to apply at the US consulate in your home country. THis will trigger a ten year bar and I don't believe there is a waiver for this (father to son). So please have a full consult done with a competent immigration attorney to fully dig into the specifics of your case. There may also be some weird claim of USC or other avenues for you, depending upon your mother's history and others.

If you do not have a petition filed for you prior to April 30, 2001, and assuming that you are over stayed your visa, you will have to go back to your country and are subject to 10 year bar.

Your permanent resident father should have filed the first step (I-130) for your green card as soon as he got his. At that time, you were a minor child of a permanent resident, and you would have a place in line, called a "priority date" for your green card of 2006. When he becomes a US citizen, you are a first preference, but it is still a 7 year wait. If he hasn't filed anything for you yet, it will be 7+ years. If he has filed for you it will be 2 years.