The Adventures of Super Mario Bros. 3
The first appearance of an elephant was during The Adventures of Super Mario Bros. 3 episode "7 Continents for 7 Koopas". The elephant, along with several other African animals, was transformed into a stuffed version of itself by Kooky von Koopa. However, Luigi stole Kooky's wand, and used it to turn the animals back to normal, who in turn began to chase after Kooky.
Mario is Missing!
A baby elephant was seen in Mario is Missing!. It was stolen from Nairobi National Park by Koopa Troopas. According to a tourist, the elephant's mother was "pretty mad", so Luigi had to return it quickly, proving its authenticity by answering questions about the national park.
In the same game, elephants were also mentioned in a pamphlet as one of the species gladiators fought against in the Colosseum in Ancient Rome. Also, a tourist in Nairobi reported that he "came through Mombasa and picked [his] way through lions and elephants to this capital city."
Donkey Kong Country series
In Donkey Kong Country 3: Dixie Kong's Double Trouble, an Elephant by the name of Ellie appears as an Animal Buddy in the game. Ellie returns in Donkey Kong Land III.
In Donkey Kong Country Returns and Donkey Kong Country Returns 3D, where regular animals are shown to live in Donkey Kong Island for the first time, elephants were hypnotized by the Tiki Tak Tribe into stealing DK's Banana Hoard along with the island's other animal inhabitants (Zebra, Giraffe, & Squirrel)
A small, blue-colored elephant is owned by Mona in the WarioWare series. Mona's Elephant, along other animals, help Mona when she is threatened, as being chased by the Diamond Police or The Dinosaurs.