This third song was not included in the Rudoplph TV special, I suspect, because he only wrote the music for it. The lyrics though weren't by a fellow-collaborator, but rather adapted from a poem -- albeit by another fellow, Henry Wadsworth Longfellow. It had previously been set to music in the late 18th century (by English organist John Baptiste Calkin -- don't worry, it won't be on the test), but when Marks wrote his own music for the poem in the 1950s, that has now become the version most-heard today. The song is "I Heard the Bells on Christmas Day."
(Having said that most people don't know that the music is by Johnny Marks, I suspect that almost as many people don't know that the words are by Longfellow...)
For that matter, having said that the song is not in the TV special, I should clarify by noting that it sort of is -- the music, at least. Near the very beginning of the show, as Sam the Snowman is introducing us to the tale, if you listen carefully you can hear Johnny Marks's music to "I Heard the Bells on Christmas Day" playing as underscoring.
But here's the full song. There are a lot of recordings, but since we're talking about "Rudolph," we might as well use the recording by Sam himself, Burl Ives.
Three popular Christmas songs by Johnny Marks. Not shabby.
Here's the song and afterwards, a bonus...