Hi Ryan! It's always the little things that seem to take up the most discussion!
I have to say, I would lean toward bringing the stroller into the parks. Yes, I know they're bulky, and they are a pain to get the kids out of and folded up on the bus. BUT - stroller rentals can add up. Click
HERE for the latest stroller rental prices as they are subject to change, but currently the cost of a stroller is $15.00 per day for a single passenger stroller and $31.00 per day for a double stroller. Also, you can only use them in the parks themselves. That means that if you have tired an cranky children at the end of a long day and the line for the buses is long, you'll end up carrying them.
When we had children of a similar age, we simply brought two very light umbrella strollers. I believe we purchased a plastic clip to hook the strollers together so one person could wheel them around. That also gave us the option of splitting them up if we felt like it. Plus, umbrella strollers are super easy to fold up and take on and off of the buses.
Whatever option you choose, make sure you take a bandanna, ribbon, or small toy to tie to your stroller to help you identify it. It's amazing how many people will have your exact stroller, and sometimes cast members have to move strollers to make room for other guests.
Hope that helps, and hopefully that will be your last planning disagreement.