The fire broke out following an afternoon thunderstorm in the Thai capital.
"At the time of the incident, no one was on the ferris wheel," the park said in a statement on its Facebook page. "As standard protocol, we stopped all operations of the Dino Eye before the rain, which is one of our safety measures for Dinosaur Planet."
The statement did not include details on how long it took to extinguish the fire on the ferris wheel, called the Dino Eye. It said the park was investigating the cause of the blaze.
Dinosaur Planet has been a popular attraction with residents and tourists since it opened a month ago in central Bangkok.