Ouch! It makes my eyes water just thinking of it. Have you tried trimming them instead?

I am guessing there are many more nerve recepters in the nose than elsewhere, so when you pull the hairs it causes much more pain. Proximity to the eyes could be the cause of your tears, but I don't know for sure.

Buddy Hackett, the veteran comic and frequent Johnny Carson guest- says that for acting roles and auditions that required emotional  weight- he would put his hands to his face in grief and surreptitiously yank on a nose hair. He was complimented for his sincerity and glassy eyes by many directors, he said.

I seem to remember from my anatomy that the nerve supply of the lacrimal gland: that causes tears, is the greater petrosal nerve, a division of the facial nerve. The nerve supply of the nasal lining is a branch of the trigeminal nerve. Although different nerves, they have connection through the pterygopalatine ganglion, where different sensory nerve synapse and therefore interconnect.  This ganglion is located behind the orbit, outside the skull in the ptergopalatine fissure.

Not an answer, but advice.  You can minimise the pain and the eye watering by inserting a finger and drawing it out so the hairs lie on top trapped between it and the edge of the nostril, then use tweezers to pull them out over the pad of the finger.  It gets all the ones that are long enough to show, and really lowers the pain, even makes it an interesting sensation.