I thought a prime time for contacting the parents is right after school when picking up the kid. Everyone is there waiting, so it's just natural to chit chat, esp when the kids are friends.
I'd count also those memorable school talent shows/performances and events. Another reach out avenue is volunteering, these have a higher chance to match parents with similar availability at least.
I have certainly gotten to know some parents at pick up, but there’s a whole bunch I have not met.