Well - here are your options:
Yes, you can drive to Harrison via rt 280 (about 20 min), park, and take the Path train. Many commuters do this. It's a viable option.
As far as getting a permit parking in a train town - S. Orange does have a NJ Transit lot, so you don't have to live there to get a parking permit (that's the good news)...... but the waiting list is very long (last time I called, it was 5-7 years - that's the bad news). Montclair is about a 20 min drive from Livingston, depending on where you are in Livingston, and there had been parking available in the fairly new NJ Transit lot near Montclair State University. Please visit my web site for train and bus route information links..
Unfortunately, most of the other 'train towns" only give permits to residents.
The bus is the other option. If you don't mind a short drive to the park and drive area in W. Orange, near the WO arena, you can get express busses in the morning, shortening the commute time.
Please don't hesitate to reach out to me with any other questions. I would welcome the opportunity to help you.
Good luck with your move!