Big pink riding elephant would be the more natural sequence to me if there are any non-native speakers who are curious. I know there's an established order of types of adjectives. However in this example, it's more of a part of speech order issue to my ear. With riding being both an adjective and a verb, it feels like someone named "pink" is riding a big elephant when the words are in this order. This may be due to the fact that we don't have much for case indicators. So the adjective "pink" is indistinguishable from a guy whose nickname is "Pink". "Riding" as in a riding animal vs a pack animal is identical to "riding" the verb. All you're missing is the artical "a" before big elephant and it could very well be a sentence fragment.

"Fred, riding a big elephant, burst into the arena" substitutes "pink" for a more common name and adds the article and a prepositional phrase and... some grammar that I can't put into words right now and makes it a full sentence.

I guess what I'm trying to get across is that without the adjective order being in line, it's hard to work out the sentence structure which is extremely reliant on word order in lieu of inflection.