Good day Jamie,
You DO NOT have to convert your concurrent desktop license to the Enterprise. However, you do need to have your User Types in the Enterprise. ArcGIS Desktop Standard and ArcGIS Desktop Advanced include the Parcel Fabric User Type extension that can be used with the Creator User Type when the Creator User Type is provisioned to ArcGIS Enterprise.
By default, when you purchase ArcGIS Desktop, the Creator User Type and your ArcGIS Pro license will be provisioned to your ArcGIS Online account. However, you have the option, at time of purchase, to request that the Creator User Type (with the ArcGIS Pro license) be provisioned to ArcGIS Enterprise (assuming you have an ArcGIS Enterprise license).
If the ArcGIS Desktop Creator User Type was provisioned to your ArcGIS Online subscription, you will need to call customer service to move your Creator User Type license from ArcGIS Online to ArcGIS Enterprise. When Customer Services moves the Creator User Type license, the Parcel Fabric User Type extension will also be provisioned, with the Creator User Type license, to ArcGIS Enterprise.
I will let the analyst handling your case know.