As people have said, they do exist, but are not available for purchase anymore
I believe that the original Dragon pets were removed from the store (like many other pets/wands/etc) for copyright infringement reasons. They can still be seen in game however. (as can any iitem purchased in the store pre-removal iirc, as I still have Bellatrix's "Dark Mistress" wand and the Thestry/Thestral pet)
The Midnight Dragon because it was, iirc, based off of Toothless from the How To Train Your Dragon movies
The Violet Dragon (which I have) was based off of Spyro (aka, best dragon ever), and was one of the first to be removed, before the rest of the dragon pets.
And the rest were based off of dragons from within the Harry Potter franchise, which is why they're removed due to copyright, despite their changed names.
However, WB/JK or whoever
cannot possibly claim copyright to every single dragon, given they are a mythical creature that is
legendary, has existed in stories for
thousands of years and many country has it's own version at this point... so I'd like to think that someday, dragon pets will return. Even if it is just one generic dragon pet, perhaps in a range of colours (red, yellow, green, blue for the houses?) that would be great to see them come back!
I actually made a suggestion a while back (iirc its still under review) about the possibility of the community designing pets/wands etc via competitions, if we did have something like that, a whole new dragon species could be created, much like on the site Dragon Cave (where there's dozens of dragon/wyvern/lindworm to collect and more always being created/added x3) that way there's no reason that anyone could claim copyright if its an original creation that the person who created it is willing to see in the game.
In short #BringBackDragons
Edit: would be super cool to have a rideable broom appearance/pet dragon combo too.. just sayin'