If he is scared of stairs, please request a ground floor parking when you get to the pay gate. The structure is very tall from the top and he would not like it.
The building is stable and you can only feel the "bounce" if you really concentrate on it. It is not however, pronounced.
Since you cannot be guaranteed a ground floor parking spot, I would recomend parking somewhere else and getting an uber the rest of the way. There are many places you can park for free a mile or two away and an uber should cost less than Disney parking as well. Hope you have fun.
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