Not sure of the ages of the kids (I wold say the youngest is about four or five since you used a stroller in 2012?) but if they carry a backpack to school, they can carry one in the parks. If you think about it, they would be more used to a backpack than their parents. They can then carry whatever they need/want in the parks.
For mom and dad, who probably are not used to a backpack, I suggest getting one at Wal Mart but each getting a smaller one and then do the same as the kids.
I would also suggest that before you leave home, the five of you determine what you really need in the parks and lay it all out on a bed. Then look over everything and make sure it's what you *nned* and it's just not there "in case". If you have "in case" things, pack those in one backpack and store it in a locker so you aren't carting around "in case" things. Then go through what is left and figure out how many backpacks will be needed and, more importantly, who will be carrying them.
As others have mentioned, bags of all kinds are permitted on Disney rides. What I would suggest is to take a large Ziploc (or similar) bag for each person to store things in case of rain or potential water on a ride. You can also use those for leftover food and the bags fit easily into an outside pocket of a backpack.