Update:
I went to remove the tail light to inspect the pins and didn't see anything out of the ordinary. I did reference my other tail light and noticed the 3 prong connector that goes into the brake/parking light was plugged in the reverse orientation. I guess they just mashed it in, I'm surprised the brake light functioned with the female connector plugged in backwards. Anyhow, brake/parking lights in the tails are now working. I'm debating if I even want to continue troubleshooting the headlight parking lights as it kind of looks cleaner with them always off.