No, neither is illegal. There are three species of parrots that are considered 'potentially dangerous': quakers, Patagonians and IRNs but all you need is a special permit that, among other things, says that you cannot breed them. IRNs and Patagonians would not make it on their own but Quakers survive in NJ without a problem. We have several feral colonies [see this:
https://njbigyear.wordpress.com/tag/monk-parakeet/].
About 6 years ago, my husband and I went to pick up something [can't remember what] in Fort Lee, NJ [it's next to the George Washington Bridge] and, as soon as I opened the door to the car, I heard very familiar screams going right over my head. I immediately looked up and saw six quakers flying over my head and screaming the way they always do when they fly - I teared up and almost started crying like a baby right then and there on the street from the intense feeling of nostalgia mixed with joy their screams brought me! I was, once more, a child and back home!!! You see, I grew up hearing their screams every day of my life and, as with everything else that it's always present in your life, they disappeared into the background and never really consciously registered while I was growing up but hearing them years and years later, they did not only brought me back to my childhood, I also realized how much I had been missing their screams... I am SOOOO looking forward to moving back home, I can hardly wait!