In the Catholic church liturgy of the time, alternating verses of the texts were sung unaccompanied by the choir, and replaced by a organ solo, either pre-composed or improvised. The congregation did not take any part in the music.
This procedure was the prescribed method in France from the 15th century until the start of the 20th century. Since printed music could only be published in France with Royal permission, most of the organ repertoire was improvised.
The notion that half of the text should be omitted from the religious service seems strange by modern standards, but official documents from the 17th century onwards indicate that the text could be spoken while the organ sections were being played, or if the organ music was based on a plainsong cantus firmus, a soloist could sing the text in unison with that part of the counterpoint.
Most likely, the recording is following some specific (and documented) contemporary performance instructions as to the chants being used, but unless the recording information identifies it, it would be hard to discover exactly what instructions were being followed.