I doubt it's a good tactic
Dropping knights will be quite surprising (and a bit funny, too ^^), but alas I doubt it's a very good idea in practice.
-We shall rain like angels of death! Oh, lord! Thy are so numerous!
An Eagle can carry 500 kg of weight, and let's say a plate-armored, paratrooper knight would weight at least 100 kg : assuming they weight naked 70 kg, add 20 kg for armor and a good 10 kg for a very efficient parachute. To keep things simple, we can say that the eagle can hold 4 paraknights for 100 kg each, and 100 kg is lost for the pilot and any harness, grip and eagle armor it should have.
You also told that eagles are as rare as elephants. I read here and there that there were 20 war elephants in some battles. Let's be very generous and take 50 of them. This means you have 4x50 = 200 droppable paraknights at once. While it's quite high, we have to remember it's a high-estimation, silk cost a lot in medieval times (making parachutes expensive), that 50 people are only there to drop knights (you lose 1/5 of your forces for your main-attack), and outside small skirmishes, battles tend to be bigger than the hundreds. All of this is worsened by...
Landing into troubles
-Stay in tight formation, my friends! Oh lord! I'm circled!
Airdrops are not very accurate and not all people will reach ground at the same time (because they jumped of or opened their parachutes at different times). Supposing they don't land outside the castle, formations cannot be set-up right after the drop : Paraknights will need time to regroup, if they can ever join their comrades. Being few is already precarious, but combining it with poor formation and preparation makes it a lot worse. Ah and yes, remember they land with the parachute?
-None shall stand before me! Oh lord! I can't see a thing!
They're likely to be blinded on landing as the parachute drops on them, I can already see knights trying desperately to remove the cloth while they're getting beaten up by the defenders ^^. It'll make landing and regrouping even harder than usual.
Narrowing the distance
-Hey there are paraknights coming.
-You know what time is it? Shooting practice!
The castle guards will have bows or crossbows. Thus, against any invader which aren't on the line of fire of friendlies will get shot. To simplify a lot1, any paraknight inside the castle isn't likely to be fired at (arrows can reach the other side of the walls, hitting other defenders), but any knight outside is sure to receive an arrow in the knee, if not the head. I know you want soldiers in the castle, not outside, but here's the problem : If there's a lot of wind, your knights will need to take some distance from the castle to land. Beyond the landing accuracy loss, this means they might be in shooting range before they get inside the castle, and risk getting killed before they even land.
Are there alternatives which still include some kind of airdrops?
I know it's not what exactly the epic siege you're hoping for, but here are a few alternatives I can think of which still includes dropping stuff on people :
- Ranged weaponry : Arrows, rocks, etc. Since you can throw/drop them from the safety of your eagle, it could cause some damage. However, if eagles are well-known in your world, castle lords will build roofs for towers and remparts to protect against them, or use their own eagles for defense. This could complicate things quickly.
- Some rancid cow corpses (and other animals) : It's a biological warfare which was sometimes used in medieval times. The advantage of eagles over traditional catapult illness throwing is that landing in the castle is (mostly) assured. And if you manage to aim the well (supposing it has no roof again), they're really going to need a good doctor.
Hopefully it could help devising new strategies for your siege :).
1 : There are also cases archers would be able to safely shoot towards the inside of the castle, like when there are roofs, or when there's no soldier on the other side. Having little to no soldier is possible if the other side is against a cliff, or with some coordination if the other side waits knights to land before going out and attack them.