New battery- car only starts in neutral

Have a 02 Mazda protege. Had no problems with it other than needing an alignment. I even drove it 3 hours away and back with no issues. A week after this, it seemed the battery died. Took it to get charged and was told it had a bad cell. I put it back in the car and alarm sounded and wouldn't turn off so I just disconnected it til I had the extra money for a decent battery. Fast forward to now. I bought a brand new battery, put it in, alarm sounded, but turned off. Lights on dash work, radio work, power windows etc work. No crank. Today, I went out in the car and tried starting it in neutral and zoom zoom! It started right up! Turned car off and tried in park again, nothing. Went back to neutral and started right away! Even switched gears and it didn't die or anything on me. I was told by one friend it might be the neutral safety switch, but another friend told me automatics don't have the NSS. So I'm confused. I don't have a lot of cash right now so I'm hoping I can either drive it like it is until after CHRISTMAS, or hopefully it's a cheaper thing to fix. Also what's the part called? Or part number if you know it. Also, how much is the average cost to fix this?