Since the network analysis is not concerned with specific attributes or properties of the features, only their location on and relation to the network, there isn't really a way to include them as part of the solver output.
However, you can take your polygon results, export them to a standalone shapefile/feature class, and then use a join, spatial join, or other method (centroids, intersects, etc) to transfer attributes from the parcel to the service area polygons.
Depending on how many inputs you had (and maybe even just one), there may be attributes that identify the service area polygon as sourced to a specific parcel/input that could then be used in a regular (rather than spatial) join. Service polygons should have a FacilityID and possibly Name attribute that links to the Facilities, and Facilities should have a Name attribute that either matches to a Name field or the OID field of the source. Note you may have to enable their inclusion in the output, ensure there is a Name field present in the source, and/or properly map that field in the solver configuration. See the help files for more details.