When you Export your board design to a ParaSolid (viewable in SOLIDWORKS) using File ► Export ► PARASOLID, The export tool will first ask you for a directory location, and will then display the Export Options panel. More details on this panel and how it works can be found in our Documentation here: https://www.altium.com/documentation/altium-designer/workspacemanager-dlg-exportsteppropertiesformexport-options-ad
This panel gives you the ability to select Copper Export Options - make sure the Export Without Copper selection is disabled if you wish the Copper to be exported as well.
If you select "Export All", you should find your copper traces will be included along with your bare board and components.
For an additional, more collaborative way to translate between ECAD and MCAD, check out MCAD CoDesigner .