Polygons would not be a good representation of the data in this case because the geometry of the parcels is clipped to the boundary of the tiles, so the area, perimeter, and parcel boundaries of the properties on the edge of the tile would be misleading. A closed polyline indicates the entire property is included in the tile, an open polyline indicates that only a portion of the property is included in that tile, the remaining area will appear in the adjacent tile. There are several methods to convert the geometry, a few are discussed below:

In CAD:
Copy the line work from one file, and use the command “paste to original coordinates” to have them together in one DWG. Use the command PEDIT (polyline edit). Select M (multiple), and choose all the linework from both tiles and hit enter. You will be prompted with, “Convert Lines and Arcs to polylines?” choose Y (yes), then enter the option J (join), the join type is “extend”. Enter a small fuzz distance 5 (mm). The line work from adjacent tiles will be joined into single line features.

In ArcMap:
To convert a polyline tile to polygons with an “Editor” license, in ArcToolbox, export the polyline file as text using “Write Features to Text File”. Open the *.txt file you’ve created and edit the header information to read “polygon” instead of “polyline.” In ArcToolbox, import the modified text file using “Create Features from Text File”. A polygon shape file will be created.
or:
In ArcCatalog, right-click the line layer, export it as a *.dxf file. In ArcMap load this * .dxf file and it will behave like a DWG file, comprised of several layers, including a polygon file. Export the polygon file and point file and save as a shapefile. Choose the matching portions of parcels which cross the tile boundaries and merge the features. Use the “transfer attribute” tools or “spatial join” tools to transfer PIN attributes from the point file to the polygon file based on spatial location.
or:
Using a script, an example can be found at the link.
or:
Using a utility such GeoWizard, available here.