Repertoire.BMI.com lists 4 different songs (or tunes) called ON MY FATHER'S SIDE:

One was written by John L. Reinhardt, Jr., and published by Chestnut Mound Music.
One was written by Felecia Shiflett and published by Felecia Shiflett Music.
One was written by Louie Nettles and published by Love Peace and Happiness Music.
One was written by Gary W. Duty and published by Carried Away Music.

In addition, Repertoire.BMI.com lists a song called ON MY MOTHER'S SIDE, which is credited to Jan Buckner, Felecia Shiflett, and John Turner. It was published by Canaanland Music and by Felecia Shiflett Music.

...and a song called MY FATHER'S SIDE, which written by Harry Reynolds and published by Universal Songs of Polygram International, Inc.

The U.S. Copyright Office says Reinhardt's song ON MY FATHER'S SIDE was published as an audio cassette in 1990.

Allmusic.com says that Shiflett's song ON MY FATHER'S SIDE appeared on 2 albums by the Village Singers: "There's a Great Day Coming" (1999) and "When I Get There" (2002).

The Reinhardt song, sung by The Reinhardts, can be seen and heard in a video at YouTube. The lyrics that they sing are very close to the lyrics that I copied from the Southern Gospel News web site and posted above.

YouTube has recordings of the Shiflett song sung by
The Village Singers,
Liberty,
The Barn Again Gang [sic],
and many others. The words are close to those posted by Grover above.

Judging by the number of amateur recordings posted a YouTube (which I have not listed), I'd say the Shiflett song is far more popular than the Reinhardt song.

I haven't found any recordings of the songs by Nettles, Duty, or Reynolds.