[stuartnz]

I'm not a teacher, but I would say that you take a picture "of" an elephant. I think most native English speakers would understand a picture "from" an elephant to mean a photo that was taken while the photographer was sitting on an elephant. Or, if you said "I took a photo from an elephant", it could easily be understood as meaning that the elephant was holding the picture and you took it back.

You could say, "an elephant's picture", but I think the most common construction would likely be " a picture of an elephant".