These answers are really interesting! I think the custom of having parties at a "place," where the time is limited (usually it's 1-2 hours), makes it really hard to open presents at the party. When we have parties at home, we open gifts; when they are at a rental place, we don't. Now I feel terrible that people will think it's rude. We had my son's birthday party at a children's museum last week and only had the party room for 1 hour. By the time food/cake was done there was 15 minutes left, and another group waiting, so we decided to skip the presents.
But 4 hours is a long time, and a 1yo party is more for the grownups than the kids, so why not do the opening there, and whoever isn't interested in it can socialize/play in another area.