This song reads like a playground hand-jive, but sounds like a Beach Boys track. There is a great deal of childhood imagery... and then a whole lot of car-racing imagery. The title itself is, in fact, the two words "baby" and "driver."
The jump-rope sing-song element is the "my daddy"/"my mama" part. The information about the parents is adult, however, and at least somewhat autobiographical. Simon's father was a very successful session bass-player, for instance. In one interview, Simon recalls a song coming on the radio and his father off-handedly remarking, "I think I played on that." (I admit I have no idea if any of the military information has any basis in fact.)
Another childhood element is the phrase "once upon," as "once upon a time." Yet another is the invitation "come to my room and play."
(The speaker does mention the circumstances of his birth, but that can hardly be counted as childhood imagery. Many songs have lyrics like "born in the USA" or "born to be wild.")
As for racing imagery, there is the chorus, which mentions "wheels," the "road," an "engine," and the line "what's my number," as all racecars have numbers.
Whether childhood imagery or car imagery, by the end of the song, they both seem to be metaphors for sex. "I wonder how your engine feels" refers to the same thing as the line in Springsteen's "Born to Run": "strap your hands 'cross my engines."
And then there is the blatant line: "Yes we can play/ I'm not talkin' 'bout your pigtails/ I was talkin' 'bout your sex appeal."
My theory? It's about a guy trying to lose his virginity. Put together, the song seems to be one giant come-on. He is young, still a "baby," with no accomplishments to his name, so he brags about his parents as a way of strutting.
Further, he "wonders how [the girl's] engine feels", and wants to "play," but has as much intention of staying around as Dion's Wanderer: "I hit the road and I'm gone... scoot down the road..." (The Wanderer explains, "When I find myself falling for some girl/ I hop right into my car and I drive around the world.")
The line "What's my number?" could then mean "You don't even know my phone number or address, do you? I'm gone before you can find out."
The virginity theory also explains the line about carrying a "gun," but not yet getting a chance to "serve"-- i.e. use his "gun" to serve anyone else.
He is a "baby driver," with temporary tags and a learner's permit, but still no license. This would explain his ridiculous attempts at seduction... and his likelihood of crashing instead of making it all the way around the track.
Next Song: The Only Living Boy in New York