Ponies have a society that resembles a meld of pre-1900s America, and victorian England. At that point in our timeline, there were less that 1 billion humans on the planet, and ponies would logically have less than that if there are also dragons, minotaurs, zebras, buffalo, etc. On their planet. I estimate there to be 100-200 million ponies, which means that even if nations like N. Korea and Cuba refuse to help, allied humans would still outnumber ponies by factors greater than 25:1. If all else failed, we could simply outf*ck ponies. We're used to our enviroment being suboptimal, a war of attrition would heavily favor humans.

That being said, this only applies if ponies invaded earth, because the converse wouldn't happen. All of the larger countries would be more interested in selling stuff to the ponies rather than killing them, and all smaller countries would be too terrified of retribution from the larger countries to invade.