Yes, it does seem to be a parody of SHIFTING WHISPERING SANDS
Here's my transcription from the recording at YouTube:
[I have not attempted to reproduce the Scandinavian accent. (Despite living in Minnesota for 40 years, I have never learned to distinguish between Swedish and Norwegian accents; they sound alike to me.)]
DRIFTING VISTLING SNOW As sung by Slim Jim and the Westerners with the Honeytones.
('Twas a cold December evening and the wind began to blow, And soon ev'rything was covered by the drifting whistling snow.)
[SPOKEN:] I found the valley of the drifting whistling snow between two great big snow-banks when I opened my door next morning, and for hours and hours I wandered aimlessly through the snow seeking the answers to the many questions that was racing through my feeble brain: Where was everything? Where was the sidewalk? Where was my driveway? My ... (?) At last I was all pooped out, so I took a chew of snus* and stretched my head and wondered where to go. Then all at once I realized I was a prisoner here in the valley of the drifting whistling snow.
[SUNG:] And my wife she whispered to me: "I got to go to Ladies' Aid. Shovel out that old jalopy." And she handed me the spade.
Now the settlers and the miners fought those crazy Navajos, But I tell you that was nothing like the drifting whistling snow.
So I took my scoop and started where that old jalopy stood, And after endless hours of shoveling snow, all I saw was just the hood.
Then the snow began to whistling and the wind began to blow, And my jalopy still is resting neath the drifting whistling snow.
Now I left my old snow shovel lying there upon the snow While I went in the house to dry my hands and warm my frozen toes.
While I was gone, the snow-bank began to grow and grow. Now my shovel too is resting neath the drifting whistling snow. (The drifting whistling snow.)