Reading the standard Nidoking route guide, I wonder why Paras is the only other Pokemon caught. I get that it's for Cut, of course, but why specifically Paras? Dig can also be taught to Squirtle and Cut could be taught to Farfetch'd. Or did I overlook something?

Its the perfect cut/dig pokemon and you get it in a yoloball at the end of mt moon manip. Getting farfetchd requires trading spearow, but advance routes catch pidgey with a yolo ball on extended manip. Pidgey cry is also faster, and the trade is very very slow. Only other viable option is oddish but that would require doing another manip which is slow due to extra save and quit, when you can just get paras at end of moon manip

Fair enough. I'd actually not taken into consideration the length of Pokemon cries to the speedrun, a minor thing but yeah that'll add up if you have to switch it in for confusion several times.