Do you have your street data (ie, you're not working in ArcGIS Online)? If so, try this:
- Add the network dataset's junction feature class to the map. This is the automatically-generated feature class of network junctions (like intersections) that lives in the same feature dataset as the network.
- Do a Select by Location and select all junctions within some very small distance of the park boundaries.
- Export the selected junctions to a new feature class. These selected junctions can serve as a sort of proxy for park entrances. You can use them as input for your network analysis.