We were so glad that we had the following places in the parks for Caleb during the day to give him some “down time”. The parks can be very overwhelming and a can be a bit of sensory overload. So it was so helpful to have areas in the parks where we could go and get some much needed quiet time. Also I would recommend bringing whatever your child likes into the parks that keeps them calm and distracted. For Caleb he loves to look down at his tablet and have his headphones on. This also worked as a “reward” system in the parks as well.
Magic Kingdom
Tom Sawyers Island (Frontier Land)
You will take a small boat ride across to the island. Once you are here it's almost like you aren't in Magic Kingdom anymore. The island has many pathways; is covered in tree's (great shade on hot days) and has some fort areas. We enjoyed walking around and exploring and it was great for some down time from all the noise and business of the park itself.

Storybook Circus (Fantasy Land)
In the back corner right next to Pete's Silly Sideshow (indoor character photo op) you will find a large tented area with chairs, tables and also areas to charge phones or tablets too. It was very quiet back here and nice and relaxing. I've included a video below of the area for you to see.
Baby Care Center
This is located at the end of Main Street; turn to the left in the front of the green grassy area. Look for the Baby Center right before the Crystal Palace restaurant on your left hand side. They have a room inside with kid sized tables and chairs to be able to relax in. There is also an area for nursing moms with rocking chairs, as well as a large changing area, and also has some very clean restrooms inside. It is also air-conditioned, quiet and calm.

Sleepy Hollow
You will find Sleepy Hollow nestled just outside the entrance to Liberty Square. There are some picnic tables under a relatively shaded patio area (we lucked out with an area in the back corner) and even thought there are only about 5 - 6 tables or so; we found it relatively not crowded at all and a nice shaded quiet area to eat a lunch we picked up around the corner.
Nestled Area with Picnic Tables near the Restaurant Area of sleepy hollow....

Hollywood Studios
Baby Care Center
When you first enter the park to your left hand side you will find the Guest Relations building; inside this building is the Baby Care Center which is a quiet air conditioned area to cool off with your child if you need some downtime near the front of the park.

Baseline Tap House
This area is just past ABC Commissary on your way to Muppet Vision 3D. This is an area of the park that is less crowded. Outside the Tap House is a covered area with Tables & Umbrella's where you can just find a seat and feel free to sit down. There is no reservations here you can just take a seat even if you aren't getting anything. We did however walk inside and grab a pint and brought it outside. There are also restrooms right nearby and food as well. You could always do a grab and go from somewhere in the parka and come back here to eat it.

Animation Courtyard
This area is located to the left of Star Wars Launch Bay; it is very quiet back here and (not shown in picture below) are some concrete benches just beyond the gift shop (on the left hand side) tucked under the overhang. My father-in-law actually fell asleep back in this area for a quick cat nap.

Animal Kingdom
Pandora
Yes this is a newer land and can be crowded; however we were there during an extremely busy time and we felt the outdoor seating area at Satu'li Canteen made for a nice sit down meal with a nice shaded and calm atmosphere.

Asia - Maharajah Jungle Trek
This is a wildlife trail in the land of Asia. It features bats and tigers and there are many places along the path where you can stop and relax.
Asia - Gorilla Falls Exploration Trail
This is also a wildlife trail in the land of Asia where you can see Gorillas as it is appropriately named as well as Monkeys and other primates. A lot of guests usually do this right after they ride on Kilimanjaro Safaris. It is very easy to rest and relax along this path.
Africa - Discover Island Trails
If you walk behind the Tree of Life in the center of Animal Kingdom you will find a twisting trail. These walking trails include animals such as Kangaroos, Otters, Lemurs, and Porcupines. You will also find a photo op (no photo pass here; you have to take it with your own camera) with Pocahontas.

Discovery Island
The Baby Care Center & First Aid is next to the Creature Comforts. The Baby Care Center is air conditioned and has a nice little quiet area to relax with your child if you need a quiet area to get away.
Epcot
Italian Gardens
Walk away from the Pavilion (with your back to it) and head towards the Lagoon. You will find some quiet seating areas. This is the Italian Pavillion:

You will find some beautiful gardens (depending on time of year you may catch the Flower and Garden Festival) and some benches along the pathway.

United Kingdom
Find our way to the middle of the United Kingdom Pavilion; located away from the pub. There is a garden area at the back of the Pavilion. You may see some shows and characters along the way but keep heading to the back area. There is a shaded hedge maze and it is very quiet area.

Morocco
Morocco has a few different places to escape to. To the left side in Morocco you will enter a building and if you continue walking you will cross an atrium that is cool a quiet; especially for those hot summer days. In the very back of the Morocco Pavilion there is a covered area that is very quiet and cool.
Japan
There is a tranquil area if you walk into the Japan Pavilion and head to your left. There is a garden area with a Koi pond that is perfect to relax in.

Mexico
The main attraction in Mexico is the Large Pyramid that houses most of the attractions. If you walk outside to the right of the pavilion there is an area near the backstage gate; as long as Donald Duck is not doing a character meet & greet at that time it is a great quiet place to get some R & R.
First Aid / Baby Care Center
This center is the same as the other 3 parks. Inside you will find areas for nursing, changing diapers, and of course a room for the kids to have a nice quiet area to relax and get some cool air as well.

International Gateway (Back Entrance)
If you happen to be on the boardwalk or have taken a boat over to Epcot (from Hollywood Studios area / or some resorts) you would be entering in near France & United Kingdom which is a much quieter and less crowded way to enter Epcot.