"I'm not even embarrassed to hug my parents in public," suggests a comfortable and loving relationship with family members. It reflects a sense of closeness and openness, where expressing affection publicly is natural and accepted.
The speaker highlights an exception: when Nathan wears a sweatband while running, which seems to be a personal pet peeve. This adds a humorous touch, emphasizing that certain habits or moments can still be mildly embarrassing, even in a generally relaxed attitude towards public displays of affection.