On the surface, this is an incredibly creepy song about about a man with bad intentions stalking a child. Kinks frontman Ray Davies, who wrote the song, explained that it is actually about a divorced dad who only gets custody of his daughter once a week. Davies says that when the guy says, "I'm not a flasher in a rain coat, I'm not a dirty old man," we can choose to believe him.
Davies was an avid runner and spent a lot of time running through parks he noticed that on Sundays these parks were filled with part-time fathers trying to make a special day with their children. Often, the dads would try too hard and the kids would be either overwhelmed or bored.
Originally, this song contained the line, "Sunday parents with their kinds knowing they're just alone," which would have clarified the meaning of the song. Ray Davies decided to take it out in order to preserve the ambiguity in the meaning, which is a hallmark of his songwriting (see: "Lola").
Understandably, this was never released as a single, as it was clear the BBC wouldn't play it. Ray Davies told Creem: "It's a good song. It's a sad song. And I'd love it to be a single. I wouldn't care if it bombed and died a death because I believe in that song so much."