The Dragons in Yoshi's Story are gigantic, armless friendly beasts whose bodies are thin and curvy. They help the Yoshi (species) cruise across dangerous areas by letting them on their back. There's a green one in Cloud Cruising on page 3 and a red one in Magma Castle on page 6.
Green Dragon in Cloud Cruising with Light Blue Yoshi
Red Dragon in Magma Castle with Yoshi