Inspired by this parody, why didn't Frodo and the fellowship use eagles to travel (even part of the way) into Mordor?
Because Sauron's Nazgul patrols would have spotted them and easily killed the eagles. Stealth was a major part of the plan to sneak into Mordor, hence the diversion of Gandalf riding with Pippin to Minas Tirith when Sauron believes Pippin is in posession of the ring. Also, the Eagles wouldnt have wanted to. Would you like to give someone a piggy back in scale probably 300 miles?
The other answers are good. In addition, Bob the Angry Flower addresses your question here
Though obviously satiric, Bob has a good point here. The fundamental problem is not getting the Ring to the fire undetected; the problem is choosing to throw it in the fire once you're there. Only four people in history have the opportunity to do so (Sauron, Isildur, Frodo and Gollum), and none of them choose to do so.
Assuming that the eagle plan would work, all that accomplishes is it gets Frodo to the point of choosing to not destroy the ring faster, which is not helpful.
In the films, there's no reason. There are some aspects of the below that you could infer, but they are largely not stated plainly in the film.
Just as Gandalf could not take the Ring himself to Mordor, neither could the Eagles. Both were bound to not take power in Middle-earth, but rather to assist and guide, as they were agents of the Valar who ceased directly ruling Middle-earth ages prior. This is why the Eagles and Gandalf did not directly take charge of the quest, but instead contributed by guiding the success of the quest. This is part of the larger theme of the smaller people of the world rising to the responsibility of taking care of the world. The Elves were wrapping up their time in Middle-earth, the Free Peoples (less Elves) needed to start standing on their own to guide the fate of Middle-earth. The destruction of the One Ring is their crucible for this.
In addition to this, by taking the ring and/or ringbearer, the Eagles would also struggle with the desire to take the ring. This is just as Gandalf, Galadriel, Boromir, et al. who assisted Frodo but had to fight the lure of the ring.
This page has a thorough run down on many points for and against the so-called 'Eagle Plan', as had been discussed on Usenet.
Eagles seem to be a strong proud species and not one to cater to just any whim. Saving the hobbits who ended the reign of Sauron? Sure, they'll help there. Being a general taxi service? No.
There's just no good reason. Stealth and diversion were indeed reasons to keep the fellowship small, but they could just have send eagles to different directions so Sauron wouldn't know which one was carrying Frodo, they could even have eagles ferry random people between random points for weeks at a time so they wouldn't attract much attention after a while and there'd be no way to know when Frodo was transported and where to. Also even if the dragon air forcewas large enough to intercept all the eagles (if it was why did we see so few in the battles?) over, or near, Mordor then the Eagles could still have been used to travel part of the way, cutting months of the journey and avoiding many instances where the fate of the world relied on pure luck (someone not falling off a cliff for example).
Stuff happens when you write a long story over a long period of time, you lose oversight, forget to rewrite all relevant parts of the story when you change one thing, etc... and that's where plot holes come from.