You're referring to the Parallel Processing environment. As you've mentioned, this environment is only for tools. Only some tools support this environment. You'll need to look at individual tool's help page to see if they support that. Note, the only options to export a map are through the buttons in the software, or using arcpy functions, meaning there are no export tools. To the best of my knowledge, no arcpy functions support parallel processing.
In short, there is no setting or option you can enable to make ArcGIS use more than 1 core when doing a map export. You've tagged your question as arcgis-desktop, meaning you could be using ArcMap or ArcGIS Pro. At this point, its incredibly unlikely ArcMap will be enhanced to make use of multiple cores when doing an export. ArcGIS Pro may simply make use of multiple cores (again, no setting to enable or disable): it does for various operations. I've never seen a list of what operations work over multiple cores. If it doesn't for exporting, and you feel strongly, I'd suggest adding an idea to ideas.arcgis.com with this.