Possible....but you better be in great shape. Is your older child going to be walking or in a stroller too? If walking - how old is that child? Even the closest parking spots can be far at the end of a long day after walking all the way out of the park - to the monorail (they might make you take the stroller down - depending how busy and if the handicapped cars are taken - so you better take the boat) or boat, then out of the gates at the TTC...and then to your car. So, you might end up on the trams anyway unless you get lucky by being really close. We always go right for opening and I've only thought it was easier to walk rather than break down the stroller like twice ever. Every other time we've been far enough away (and tired enough after a lot of walking and a long day) to take the tram.
So, a word of advice on breaking down strollers just in case you end up deciding to do this. Bring a large, cheap, roomy canvas bag and keep it on the bottom of your stroller. This, in addition to your diaper bag or whatever other bag your 'stuff' will be in. Leave this bag in your stroller throughout the day. When you get close to a bus or tram where you need to break down the stroller - grab it out and throw anything you've accumulated in the stroller basket in it. THis is usually the biggest pain for us - we end up with some water bottles, garbage (wrappers or napkins, etc), popcorn bins, shoes or socks that were taken off to dry after running through splash pools, etc. at the bottom and yes, it CAN be a pain to scramble to try to have a hand to take all these things and hold them on the bus or tram. That's why having a large bag to take anything and everything and keep it in one place while the other adult is breaking down the stroller works great. Oh and don't worry - with 2 year old twins even if they're sleepign we find other parents hugely helpful at WDW. I had my hands full with a child and just asked another parent to break down the stroller and he did and handded it to me and everything. No problems at all!