This is kind of a complicated question on some levels, but as it turns out the answer is pretty simple.

The evolution of longevity is controlled by a number of things and there are a lot of factors that go into each individual's lifespan. But generally speaking extrinsic mortality will be the biggest driver of evolution of longevity, and as extrinsic mortality decreases the average lifespan of individuals in the population will increase. The reason for this is that lower rates of extrinsic mortality will make allow the parrots to breed longer and so selection will not weaken as quickly later in life.

And parrots seem to have a lot going for them there.

They live in really stable environments and for the most part they enjoy year-round abundance. The exception there being the granivores which have a seasonal food supply and seem to live the longest among the parrots. It's been suggested that that is the result of caloric restriction dictated by the seasonal food supply.
Source: Auk (2006)

But most importantly for the birds is that they can fly. Predators aren't as large of an issue for the flying vertebrates as they are for us lowly ground dwellers.
So in the wild parrots just seem to get to enjoy the benefits of living in a great habitat.

In captive birds it seems that "owner error" is the largest cause of death. And that ranges from not giving them enough water to giving them too much contaminated water. So they aren't even really dying of old age in captivity.

All of these things add up to a parrot that pound-for-pound out-lives all the rest of the birds.

Horses on the other hand have been evolving for many generations under artificial selection and extrinsic mortality for them is fairly high. A 2011 study found that 94% of horse death is the result of euthanasia. They get put down most often simply because they're lame.

So like I said, in the end the answer is simple.
Parrots enjoy a long life for all the reasons we might have expected and horses don't live as long as parrots because they most often die when they are put down by their owners.