I agree with the others. I've done it with one stroller as a solo parent with one toddler. It was a massive pain in the neck. If the adults in your party can handle the children, your bags and the folded stroller, you will be fine. If kids outnumber the adults, you may want to re-think your plan. I honestly don't know if it's that big of a time saver. You have to wait for the train and it also stops in Fantasyland before getting to Main St.
Edited to add: Why not just watch the parade from Frontierland?