First of all, don’t use 3D renderings as the final authority for showing how a board will be fabricated. The board fabricator will work from the Gerber files, so when you notice discrepancies like this you should verify the situation with a Gerber viewer. (Regardless of what DRC or any other tools may tell me, I always do a final review with a Gerber viewer before I send a board out for fabrication.)
To resolve this situation I would bring up the affected footprint(s) in the Footprint Editor. Look at the parameters defined for each pad. Are they specified as through-hole pads? Are the appropriate check-boxes marked for ALL affected layers? Are there any values for Solder Paste Clearance or Solder Mask Clearance, that would over-ride the global values? You may find it helpful to compare the values and parameters from a known-good footprint, to a faulty footprint.
If you make any changes be sure to save them to your footprint library - either modifying the existing footprint, or creating a new footprint. (As difficult as it is to imagine, somebody may have had a valid reason for covering those pads with soldermask, so you might want to keep the footprint around.) Then do what’s necessary to update the footprint associations on your schematic, and swap out the new and old footprints on your board.