Great topic.

I think the Stones are serious, but most of the "rock goes country" that I'm aware of is parodic (even "Sweetheart of the Rodeo" may seem a bit tongue-in-cheek). I actually sort of like both versions of "Act Naturally."

Kinks: https://youtu.be/rucD74L8fFo

Turtles (country): https://youtu.be/41u21jULCqw

Turtles (bluegrass): https://youtu.be/X3UwT2nLqaE

Bonzo Dog Band (western in the Johnny Cash / Marty Robbins vein): https://youtu.be/U1ssuB7lb98

I have a longer analysis of the Turtles' Battle of the Bands album in the shop, ideally uploaded soon.