OP here with an update. For the Labor Day weekend event, the host finally called my dad, on Thursday I think, and let him know about it that way. I never found out if my mom received the email or not. We all thought it was weird that email had been the primary method of communication. And, my parents and I ended up going to the event today.
This leads to a related etiquette question! For the event, we were supposed to meet at a specific restaurant, Chez Fancy, at 11:30am. We got there on time, only to find the host and other guests hanging out in the parking lot--due to it being Labor Day, the restaurant wasn't opening until 12:30pm, another hour. (I don't blame the host; apparently on the restaurant's website it said they would open at 11:30am, and made no mention of possible changes due to the holiday.) I said, "So, what's the plan?" The host wanted to go to a fast food place across the street, to get coffee and wait until Chez Fancy was open (about an hour). I was kind of surprised by that. I thought they would say, "Oh, there's another nice-ish restaurant down the street we were thinking of going to, which is already open."
On the one hand, they really wanted to eat at Chez Fancy for their special occasion. On the other, they've suddenly added an hour to the event for everyone, when we all had to travel about an hour to get there anyway. Luckily no one in our group had a tight schedule, so we were able to wait for Chez Fancy to open. But, I was curious what others might have done in that situation, as the host.