Am I right to assume it would be side lights and tail light that would be on in park setting and only one side of the car? If not I'll investigate that further!
You are correct, no relation to hazard flasher and turn signal lights.
You most likely have a defective flasher module. ( it's part of the big triangular switch assembly )
( I was also a defective flasher back in my youth, but let's not go there )