To start this off, I have seen hundreds of rants by fans saying that Dumbledore can use time turners to save Lily and James Potter, and, I think it is complete nonsense. Here's why:

Time Turners have always fascinated me, they seem like a plothole on the surface, but as you dig deeper, they are not. The way time travel works in Harry Potter, is that every action you do, happens because a specific series of events take place. To explain that further, I give you a paradox, postulated by the great Schrödinger himself, the grandfather paradox.

The paradox assumes a situation where you go back in time, and kill you grandfather before he marries your grandmother, and sires one of your parents. The paradox asks you, whether the action could really take place. Because, when you kill your grandad, you eliminate all chances of your birth, negating the killing. He later said that the idea itself was wrong in some ways, but never elaborated on it. I believe, that it is wrong, due to one cause, you travel, the chances are, either it will fail, or that it happens, but not at the right time. Because, you see, the reason you could go back in time, was due to a set of events allowing it to happen, trying to eliminate on of these, would not succeed, because you will not be able to go to the past to do your action.

Due to the argument above, it can be inferred, that the saving of Sirius and Buckbeak, was not altering the future, it was what happened anyway. For all you know, there might have been an attempt to do just that, and it would inevitably fail. Further, the whole concept of using Time Turners to alter the past seems inherently flawed, for the reasons stated above.