Well, I'm back, and I stand (well, sit) corrected. As clearly as I can hear it, via earphones and an Ipod, with the volume fairly high, the line is "each night at the old time's sake." Despite what I said about the pure rhyme, the sibilance is there, so it can't be "old time parade." My nice theory about what that would have meant (notice the past tense) is shot, and the only thing I can think is that "the Old Time's Sake" is, apparently, a regular event at a local dance hall, in the town of the writer's imagination.

So it's not "old time parade," and it's not "old kindness' sake" (there's no nasal sound in that syllable). Now, why it's "old time's sake," I don't know. Ask Linda Thompson.

I hope this is clear enough.

Bob