The folder name isn't added as an attribute onto placemarks read from KML directly. The folders themselves are read separately as a feature type called Folders. The Placemarks will have a format attribute added to them called kml_parent which contains an id value that will match the kml_id from one of the Folder features. To add the folder name onto the Placemarks, you can use a FeatureMerger transformer to join on those two attributes.

Here's a sample template file to show how that would look within FME: kmlgetfoldername.fmwt

Where can I find the file described above (​kmlgetfoldername.fmwt)?

Thanks @kerkhoff​! I'm attaching that file to this comment.

Thanks Laura at Safe.