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Looked a while for the parody. Did not find it, but here is John and Glen on the Smothers Bros!
Gentle On My Mind - Glen Campbell and John Hartford - YouTube

That's a great song. Gotta be quick to sing along! I remember as a kid, the Smothers Brothers did a parody. He was a regular on their show.
I still sing "knowin' that your door is always open and...