EDIT PER YOUR ANSWER: Then the answer has to be magical weapons. The difficulty will be: how magical? Too magical and it may be a slam dunk defeat -- depending on how magical the humans are. Not enough magic and you're still stuck with sparrows dropping (magical) pebbles.
I think the big thing you have to determine is: do the humans know they're fighting sparrows or not? If a human army is marching a long distance to fight a kingdom that they don't realize is run by sparrows, you have a lot of room for maneuver.
The sparrows could at first poison/taint food and water. Then begin to attack horses and wagons necessary for carrying supplies. Have a burnt-earth policy to deny the humans food via foraging.
(I really, really think you'll also need sparrows willing to die in the fight as suicide attackers. Otherwise, as I said above, you'll have to resort to the sparrows having better magic than the humans. Other suggestions like "eat leather straps", etc, would eventually fail as the humans adapted and added nets around things, put chemicals on the leather straps (suicide for the attacking bird), etc.)
The whole idea would be to make the human supply lines difficult to maintain, and to essentially terrorize the humans who would be shocked by little birds driven to kill in suicide attacks. The sparrows could make it look like the humans were going up against some crazy mage king who could drive even the smallest of animals to suicidal attacks. Imagine what will happen when they get close enough that the mage emerges and fights them directly. (Or with the help of larger animals or monsters.)
Simultaneously, the sparrows could let other human groups -- unfriendly to the kingdom attacking the sparrows -- know that the army was stretched thin and far from home. Perfect time to attack.
Can the weapons be magical? (I.e. do you have a high- or low-magical story?)
If not, can the sparrows be a lot like sparrows but actually larger? Or have larger warriors that have been specially bred for war?
Or maybe their allies could soak them in gasoline and give them tiny sparkers, so they set themselves on fire and plunge into their enemies' camps, supply wagons, etc? Maybe 10,000 flaming sparrows would do the trick.
Or perhaps the Sparrows could be loaded up with dozens of plague-bearing fleas and could plunge into their enemies' camps. Sort of a two-step, biological attack. If their enemies were medieval-level technology-wise, it would do the trick. Of course the sparrows and their human-sized allies might also have some, um, complications.
Without some edge like that, they'll be more comical than deadly. (Unless their targets are unsuspecting, in which case you might as well have them drop poison in their targets' food/drink. Especially in a low-magic world, it would be really hard to figure out that your people are being poisoned by cute little sparrows.!)