Hello, Magnolia!
It sounds like your life is full of little Mouseketeers, I'm sure you are the leader of the cutest club ever! Thanks so much for your question, I know that planning to visit the parks with young kids requires a little extra planning.
You are correct that wagons and stroller wagons are not allowed at the Disneyland Resort. I'm sorry about that, but I'm so glad you checked before your visit so you don't have an unwelcome surprise! Strollers are welcome at the parks as long as they are no larger than 31" (79 cm) in width and 52" (132 cm) in length, but I know that finding a triple stroller is no easy task.
There are double strollers available for rent just outside the theme parks and thanks to their bench seat design I have seen three kiddos squish into one now and then. But I wouldn't recommend that for a whole day, or if any of the kids are too young to sit well on their own. Sadly, it may not be possible to safely take your little crew to the parks together until they're a little bit older or you have another adult who can join you to help. I'm sorry, I know that isn't the answer you're hoping for!
Another option could be finding a Fairy Godmother or trusted babysitter to stay with one or two of the kids so that you can take turns visiting the parks with each of them. It could be really lovely to spend some solo time with each child and let them be the guide for the day. I know that takes a lot more work on your part, but at least you'd get some extra Disney time! When you do make it to the parks I highly recommend that you spend plenty of time in
Mickey's Toontown. It's the perfect place for little imaginations to explore!
I wish I could be there to help you wheel around the parks, Magnolia!
Keep moving forward!
Amanda