From what I remember, the protagonist is pretty young, maybe 13 or 14 years old. She discovers a mythical creature on a walk outside; I think it was a griffin, but I'm not entirely sure. It's tied to her discovering she has magic powers. She has to hide the creature from the rest of her non magical family. She has a little brother.
Pretty sure it would be set in the early 90s, I remember a description of the girl talking on a corded telephone. YA fantasy.