I'm pretty sure this is the one by Pinkard & Bowden (sp?), who have a string of comic songs that get occasional airplay on country stations. This one may not be a parody of another song (their "Blue Hairs Driving in My Lane" was a great parody of "Blue Eyes Crying in the Rain"). You might find it on Cowpie, which has Country/Western lyrics. There should be a link to it somewhere here on the Mudcat, but I don't know precisely where off the top of my head.

If I remember right, the chorus starts out "Don't pet the dog, it makes him think of romance..." (maybe then followed by "Just leave him alone" etc. as you wrote above).