Hi Mary!
Congratulations on your fast-approaching trip to The Happiest Place On Earth!
You have several options available to you in order to visit several parks in the same day while having a car parked on the Walt Disney World property.
If starting your day at Disney's Animal Kingdom (which is a GREAT plan since I find that the animals on
Kilimanjaro Safaris are all out roaming & best viewed early in the day!) I wouldn't recommend leaving your car parked at the Animal Kingdom parking lot for the remainder of the day. Reason being, Animal Kingdom Park generally closes earlier than all other Disney parks and there will be no internal transportation operating TO the park later in the evening.
That being said, you are indeed able to leave Animal Kingdom with your vehicle and 'Paid Parking' ticket and show it at the parking booth as you enter the Transportation and Ticket Center parking lot at Magic Kingdom to enjoy the remainder of your day. You will not have to pay a second time to park at another Walt Disney World parking lot within the same day.
Alternatively, if you'd rather just park once and then use Disney's fantastic, internal transportation system for the duration of the day, proceed first thing in the morning to the Transportation and Ticket Center and then board a bus directly to Animal Kingdom Park.
Wishing you a Magical time on your vacation Mary!
Kirsten