I just posted the lyrics from the 1926 and 1938 versions of this but, naturally, since I did not control-A control-C before hitting "submit," the message vanished into electronic limbo.

It's a mix of straight and parody. Key verse is the opener, same in both versions exc that in 1938 the cow is only NINE miles away:

In the good old summertime
It was on the Beeno [sic; B&O, probably] Line
Stuck my head out at the window
To kiss that baby of mine
The train it ran so quick and fast
I missed that baby of mine
I kissed a cow ten miles away
In the good old summertime.

Second verse in 1926 is "Good Old Summertime" as you would have heard it on Sing Along With Mitch in 1959, the verse everybody knows.

Second verse in 1938 is a little different

In the good old summertime
In the good old summertime
Strolling down that shady lane
With that baby of mine
Oh heart to heart and hand to hand
Life is a beautiful rhyme
Go walking together, talk to one another
In the good old summertime.

Third verse in 1926

In the good old summertime
Down Broad Street after nine
If you meet a girl dressed to kill
Oh that's my baby mine
But if she gives you the telegraph wink
That is a very good sign
That she'll be your tootsie-wootsie
In the good old summertime.

The 1938 version is somewhat different:

In the good old summertime
Down Broad Street after nine
If you meet a real pretty girl
Oh, that's my baby mine
But if she gives you a pleasant look
That's a very good sign
That she will be your tootsie-wootsie
In the good old summertime.

Both 1926 and 1938 repeat the second verse to conclude the song.