Moving back to NYC

Hi, my wife and I lived in NYC in 2010 and 2011 until we had our first children and moved back to Texas to be close to family. We now have two children who are 3 and 1 and would like to move back to NYC as we miss it and think the cultural opportunities of raising our kids in such a city our worth the sacrifices. Our budget for housing would be under $4500 a month and we would be looking for a two bedroom/2 bath apartment. Previously we lived on the UWS and loved it but was wondering if there are any other neighborhoods people would recommend based on our budget. We loved being near the park, walking around, etc. Any help would be much appreciated!