They're not as exotic as some worlds' mounts, but i have gnomes that ride small cats and flying squirrels.
Worldbuilding
Rules of !Worldbuilding:
See here for a longer, more explanatory version.
- Rule 0: These are guidelines, not laws.
 - Rule 1: Be polite and respectful to others.
 - Rule 2: Provide some lore with your submissions!
 - Rule 3: Show some effort.
 - Rule 4: Do it yourself.
 - Rule 5: Advertising is limited.
 
Related Communities
For conlang (constructed languages) discussion check out !conlangs@mander.xyz Feel free to discuss the your conlangs in our community, as well!
Big ol salamanders/lizards
Salamanders are amphibians, or do you mean the mythological salamander? If they're amphibians, that would present some interesting logistic issues. Their skin would need to stay moist and they would need to lay their eggs in water.
For me, big salamanders just need a bath every day to stay healthy. I also have magic, so if a salamander is magically alligned they need exposure to a lot of that element once a day. I have them lay eggs in terrain befitting their element, so water salamanders need to lay eggs underwater but earth salamanders bury them underground.
Then theres the lizard lizards~ scaled up versions of whatever lizard suits your fancy. They need to eat a lot, but can climb walls while carrying heavy loads! Also can hold their breath for a long time and are capable of "galloping" over extremely rough terrain, unlike equivalent equestrian escorts.
For ground runners, you probably want a big terror bird. These guys are the closest to modern day dinosaurs. They have long, muscular legs that are well adapted to long distance running and even extant modern examples in the mundane world can carry impressive loads for a bird. Case in point: Children and small people can literally ride an ostrich. They have small cranial masses for their size and they're not very smart compared to a parrot or a corvid, but probably still smart enough to be tamed. They also maintain a consistent posture which is probably beneficial for a riding animal. Parrots have a flying stance and a perching stance that are nearly perpendicular to each other, so as soon as your big Macaw landed it'd dump its rider off of its back.
With the ecological niche that they fill, parrots are tree-dwelling perching birds that are not adapted well to long distance flying or ground travel. Parrots are climbers; they're for hanging around in the trees eating fruit. And their feet are for grasping, not walking. If your bird is evolved to be anything similar, trying to ride one would be rather like trying to throw a saddle on a monkey. Potentially hilarious, but unlikely to be productive.
My description was rather poor in hindsight. Their parrot-ness is mostly in the colorful feathers, large beak designed for crushing, and long macaw-like tail feathers. They're not zygodactyl like parrots, and probably have longer legs like an ostrich. I'm not sure where they exist on the food chain, but they're likely social as that would make them easier to tame. They are clever and possibly tool-users in the vein of corvids (or indeed parrots).