Helen Clements and her children Charlie and George told ITV News that her car had "overheated", causing the family to abandon the vehicle at a lion enclosure at Longleat Safari Park.
Ms Clements said: "What we thought we better do is, because what you are supposed to do is sound your horn, so we did. But my son George thought he better get out the car, so he opened the door, and got out the car. And I just said "George don't get out of the car"".
Laura Jeffries was a visitor at Longleat yesterday and saw the fire. She sent us this video.
The spokesman for Longleat Safari Park said that the visitors involved in the incident were believed to be a mother and her son and daughter who were aged around 12 or 13. Visitors said the lions were 150 metres from the people carrier when the incident happened and the car was left "burnt out".
The family remained in the smoking vehicle until the rangers arrived, and the car caught fire when they had left the area. Longleat staff said at no point were the mother and children waiting outside their vehicle.
Cars were cleared from the area following the fire to make way for the fire service, but the park was up and running again shortly after.