No. Your permit, permits you to drive in California with a parent, it is not valid in New Jersey. I wouldn t want to drive there any damn way, people drive crazy there. I especially would not want to drive there this time of year with snow, black ice and sleet on the roads being a very inexperienced driver. I have had my license over 10 years, learned to drive in the snow and am still petrified driving in it.
Most states recognize the permits issued by other states, and I d say that you definitely need a licensed driver over 21 with you (a parent qualifies!). I do not know for certain that NJ does, however. AND, the accompanying, over 21 driver, may have to hold a NJ license in order for it all to be legal.More specific information about out of state permit issues can probably be found here, please take a look:http://www.state.nj.us/mvc/.The best place to get an answer, without a doubt, is to call one of the New Jersey DMV offices and ask them about the rules.
My instinct is that you could temporarily drive there on a visit, but that if you re moving to NJ, then you should sign up for their permit program.