There's one big difference between "third party" and "third parties". One entity can be brought down and it itself has full control over the system. But thousands of small entities scattered across the whole globe can't be taken down, and even if someone started taking them down, new ones would emerge to replace them. Another key detail is that they are small - if Bitcoin required expensive server equipment to run, there would be much less nodes, because nodes don't make money, only miners do.