I would like to find the distance from Survey respondents' homes (points) to their nearest park (polyline or polygon), is there an easy way of doing this? I've used network analyst previously but only from point to point.
You can take your park polygons and turn them into polylines using the 'Polygon to Line' geoprocessing tool (this will simply take the rings of each polygon as a polyline).
Then do an intersect of your polyline features against your street features in your network dataset using the 'Intersect' geoprocessing tool (make sure to specify the output feature class type as a point feature class). This new point feature class will effectively represent all of the possible entry points into your parks along the network dataset.
Create a closest facility layer, then load your survey respondents' homes as Incidents, and load your newly created "park entry points" as Facilities. Solve to find the closest facility.