In my book series, some of the people of the planet Aurea use giant vultures called Argentavis for transport and aerial combat. These domesticated, selectively bred birds are far larger than their wild couterparts, at around 10 ft tall at the shoulder, 13 ft long, and with a 40-foot wingspan. These birds, being as large as they are, prefer soaring at extremely high altitudes to actual flight, sometimes reaching altitudes of 30k ft above sea level or more. The people of Aurea only have an early Renaissance tech level, so how could they survive riding these creatures without either freezing to death at the high altitude or suffocating from lack of oxygen?
Simple solution: Aurea's atmosphere, when compared to Earth's one, is more dense, thus at the quote of 30k feet on Aurea the air density and oxygen content are equivalent to the Earth's 18k feet: one feels it, but can still get acclimated, which is something they do.
For getting protection from the cold, don't forget that many populations lived in harshly cold conditions way before somebody brought them so called modern means of staying warm: think of Inuit, Sami, Ainus and so on: furs can do their job in protecting someone from cold.
an early Renaissance tech level
Though it would be slightly anachronistic by real-world standards, you could actually make use of a chemical oxygen generator or rebreather.
According to wikipedia's entry on the history of rebreathers:
Around 1620, in England, Cornelius Drebbel made an early oar-powered submarine. To re-oxygenate the air inside it, he likely generated oxygen by heating saltpetre (potassium nitrate) in a metal pan to emit oxygen. Heating turns the saltpetre into potassium oxide or hydroxide, which absorbs carbon dioxide from the air. That may explain why Drebbel's men were not affected by carbon dioxide build-up as much as would be expected. If so, he accidentally made a crude rebreather more than two centuries before Saint Simon Sicard's patent
Though this is clearly late in a fairly expansive definition of the renaissance, it isn't implausible that the trick could have been devised by a clever chemist (or alchemist) before then. Certainly, gunpowder technology was widespread in the real world by even quite generous definitions of "early renaissance" so access to saltpetre would not be anachronistic.
Engineering even a fairly simple rebreather isn't an entirely trivial exercise, but not an implausible one. They're also safer to build and use if you're not diving and in need of high-pressure air.
As for keeping warm... furs might be good, but also consider the benefits of simple things like hot waterbottles or suitably insulated containers with smouldering coals in. If the passengers are mostly sitting still, these make good additional choices, depending on the loadbearing capabilities of the bird in question.
Note that whilst down makes excellent modern insulation, making the best use of down requires very light fabrics with a very close weave, something that's considerably more awkward and expensive to source as you go back in history. This may or be important, depending on the economic status of the bird-riders. Fur is obviously much easier to make use of, but it is also heavier which may be problematic depending on how strong the birds are.
30000 feet is not a problem for survival, in the short term.
It is just a bit higher than the peak of Mount Everest.(29030ft). 30000ft is only 6% less pressure.
People have not only survived, but actively climbed the mountain to get to the peak of Everest without any oxygen equipment. The human body is quite capable of surviving this environment with minimal exertion, but it does require extensive training to do so, especially if any level of physical exertion is required.
Your riders will be physically completely inert, just strapped in to their mounts.
They will, of course, have to be acclimatized to mountain air.
They will need to be wearing what amounts to arctic wear. Parkas or equivalent.
They will need a period of recovery after a flight at that altitude.
It would not be a good idea to spend a cumulative total of too many hours at altitude. Any time spent above about 26000ft does do permanent damage to the body, regardless of acclimatization one cannot indefinitely survive above that altitude.