We brought our stroller when we went to WDW. It helped when walking through the airport, especially on the five mile trek through LAX at 11pm! We gate checked it both times, never had a problem.
Yes there is designated stroller parking. Beware, sometimes the CM rearrange the strollers so you may have to look for yours. Also, don't leave valuables in your stroller. I do leave my backpack, soft cooler and sweatshirts and have never had a problem and I go almost weekly. But I have heard of others not being as lucky.