Here is a list of the companion/family bathrooms. If you have a child who needs assistance in the restroom, this restroom is available to you as well.
The chances of your being near a baby care center when your child needs to use the restroom are very slim considering there is only one per park and struggling to make it across a crowded park with a small child trying to "hold it" is not only unreasonably stressful on the child, but the parents as well.
Animal Kingdom
Discovery Island opposite Flame Tree Barbecue, just before DinoLand USA entrance on the right
Harambe Village in the Mombasa Marketplace
Chester and Hester's Dinosaur Treasures in Dinoland USA
Near Maharajah Jungle Trek on the left before the entrance
Inside Maharajah Jungle Trek to the right as you exit the Bat House
Conservation Station at Rafiki's Planet Watch
DHS
Opposite Star Tours
Right side of Millionaire building on Mickey Avenue
Rock 'N' Roller Coaster
Opposite The Twilight Zone Tower of Terror
Fantasmic! amphitheater
Right side of Lights, Motors, Action!
Epcot
Spaceship Earth On Left as you face SE
Spaceship Earth on Right as you face SE near Gateway Gifts
Future World West opposite The Land - Photo 1 -- Photo 2
Future World East opposite Test Track
Refreshment Port (right side of World Showcase - before Canada)
Morocco on the right
Germany on the right
Norway opposite the Viking ship
Magic Kingdom
Lower level of Cinderella's Royal Table in the Castle
Pirates of the Caribbean
Near Splash Mountain
To the right of Space Mountain
Mickey's Toontown Fair
Transportation and Ticket Center East Gate
Baby Care Center Locations
Magic Kingdom® Park:
Only one location
The Baby Care Center is next to The Crystal Palace restaurant on Main Street, U.S.A.
Epcot:
Only one location
The Baby Care Center is inside Future World's Odyssey Center.
Disney's Hollywood Studios:
Only one location
The Baby Care Center is in the Guest Services Building inside the Main Entrance.
Disney's Animal Kingdom® Theme Park:
Only one location
The Baby Care Center is behind Creature Comforts in Discovery Island.
Disney's Typhoon Lagoon Water Park:
There is no baby Care Center, but the restrooms do have baby changing stations.
Disney's Blizzard Beach Water Park:
There is no baby Care Center, but the restrooms do have baby changing stations.
Downtown Disney® area:
There is no Baby Care Center, but the restrooms do have baby changing stations.