To make sure the rungs are parallel and square, I think you must use a square to make sure the bottom of both rails of the ladder are exactly even. Given the care you've taken to get the steps parallel on each side, that should give you a nice, square result.
Long working time, slow setting time on the glue!!!!! You don't want to be rushed unnecessarily during glue up. Trial fit everything first without glue to spot anything that will give you problems once glue is applied.
Stick showed a picture of some steel, right angle braces that are maybe 2 inches long on each side. These will be out of the way and all but invisible, or, you can paint the hardware black so they become a feature instead of just hiding them. As stick said, this will be VERY strong and will also prevent wracking. If you use these, make sure you use sizable sood screws that fit tight in the holes in the hardware. I'd use star drive type heads so you're less likely to scar the wood if you slip, which is more likely with phillips head screws. You could also use hex head screws which would all but eliminate damage from driver slips.
I would spray paint the angled hardware and screw/bolt heads before assembly and touch them up if needed.
I mentioned the height issue if someone falls. There is a formula that states a 1 story fall, landing on your feet will cause some shock and minor injuries--more serious if the child doesn't land on their feet. From two stories, a fall will often cause severe joint damage, or if the landing is not on your feet, you riks breaking bones, or hitting your head or other vital spots on objects below--a bike or tricycle for example. From 3 stories, something is going to break, period. Four stories often is fatal and five is nearly always fatal. If a child parks their tricycle near the bottom of the ladder, a slip or fall could easily change that child's life forever, if they fall onto the bike. Same applies to any restraining structures that could be in the path of the fall.
Another safety feature, to me, would be some friction type tape on the rungs of the ladder. It's pretty cheap and HD has it. It looks nice too. Certainly a painted or slick finish material will increase the chance some over excited kid misses a step, slips, and falls. If that happens, even with only minor injuries, mom will have you over immediately to remove it.
Don't mean to harp on this safety thing, but as with all things, the devil is in the details. You have pre planned really well on the construction and really, that is a beautiful project and the kids will surely love it. As a kid, you couldn't have kept me off it.
Love this picture, and it's similar to what you're building, but imagine a couple of bikes below and kids running around, making a ruckus that distracts the climber.
The more I do, the less I accomplish.