Posting links to previous answers is not intended to discourage questions or further discussion.
From /u/Elm11's reply the last time this was asked, you might like to see
The answer depends on how we define a paper airplane. Let's first consider the strictest possible definition of a paper airplane: a toy glider made only out of folded paper. The technology of papermaking has its origins in East Asia, so a good place to start is examine the history of Asian paper folding. However, we run quickly run into the problem that very little research has actually been done on topic of paper folding.
One of the few detailed accounts of the history of Chinese paper folding is presented in Tsien Tsuen-hsuin's Paper and Printing, published in 1985 as part of Joseph Needham's seminal Science and Civilisation in China series. According to Tsien, the practice of paper folding (zhezhi) emerged in the Tang dynasty at the latest. This is based on decorative folded and cut paper flowers dated to the 9th-10th centuries, which are believed to be among the oldest surviving examples of paper folding as an artform. The paper flowers were discovered by Aurel Stein at the turn of the 20th century in Dunhuang, a city in western China, and are currently held by the British Museum. However, there are no mentions made of folded paper gliders. Tsien does discuss other paper toys, but we'll come to that later.
Then what about paper folding in Japan? While the Japanese term origami has come to encompass all forms of paper folding, it may come as a surprise that research into the history of Japanese paper folding is even more lacking. Most of the research into the origins of origami comes from hobbyists and amateurs. This includes Koshiro Hatori's "History of Origami in the East and the West before Interfusion" published in 2011 and Isao Honda's The World of Origami published in 1965. Hatori traces the origins of origami to 14th century ceremonial paper folding such as noshi or ocho-mecho. These were used as wrappers; noshi derived from noshi-awabi containers for abalone meat and the ocho-mecho paper butterflies were meant to adorn bottles of alcohol. It doesn't take a huge leap to imagine that one of these early origami designs was capable of gliding through the air, even if accidentally, but that was not their intended purpose.
We've apparently reached dead end. So let's use a looser definition for a paper airplane: a toy or device made of paper (but not necessarily just paper) that is capable of flight. There are several military devices that fit this description. /u/cthulhushrugged has already discussed the shen huo fei ya ("magic-fire flying crow"), which was a bird-shaped bomb with a body made from bamboo that was covered in paper using an adhesive. Placed under the wings of the artificial bird were arrows propelled using gunpowder.
Outside of warfare, there are several pre-modern flying paper toys that are worth mentioning. Flying kites made of paper were used by the Chinese in circa 549 at the latest, when there is an account of a paper kite being used for signalling during a rescue attempt. The first mention of flying a kite as a recreational activity (and not for military or measurement purposes) comes much later in the 10th century, from a passage in the Xun chu lu that describes a person attaching a kite to a bamboo to produce music using moving air currents. These kites were designed with the paper covering a bamboo frame. Prior to the invention of paper, materials such as silk would have likely been used instead. There were even kites made large enough to supposedly lift a person.
In China, there are also paper lanterns that fly by heating the air inside the lantern with a candle, similar to how a hot air balloon works. The history of these flying tian deng lanterns is murky at best, but (non-flying) paper lanterns were popular at least by the Tang dynasty. This dating is based on a book discovered in Khotan that references paper being bought for the purpose of making lanterns.
But why limit ourselves to paper? We might as well adopt an even more flexible definition: Any toys capable of flight. /u/AshkenazeeYankee mentioned George Cayley, whose "rotary waft" experiments were inspired by the "Chinese tops" popular in 18th century Europe. In China, these tops were made from bamboo rather than folded paper. They were "helicopter tops" that would fly off after spinning their axis by hand. The origins of this toy are unknown, but the connection of the helicopter top to the Mongol prayer-wheel and Chinese zoetrope has been speculated by Needham.
Going back to our strict definition of paper airplanes—toy gliders made of folded paper—we don't really starting seeing literary references to these types of paper airplanes until the 19th century. /u/AshkenazeeYankee has already cited Charles Matthews' memoirs from 1860. There's also a line about a "broker [who] cannot... innocently fling a ‘paper dart’ at a neighbour without being amerced ten dollars" from James Knowles Medbery's Men and Mysteries of Wall Street from 1870. These "paper darts" predate the invention of airplane. However, this does not necessarily imply that there were no premodern paper airplanes that are even older; that may just be a result of the lack of research done on the history of paper folding.
A story in The British Essayists published in 1803:
...he presented himself to the wondering eyes of Euphorion with a huge black bush wig stuck full of paper darts, and as thickly spiked as the back of a porcupine.
And for reference, here is a diagram of paper darts from Cassell's Complete Book of Sports and Pastimes: Being a Compendium of Out-Door and In-Door Amusements, published in 1896. They are very similar, if not identical to a paper airplane.
Thanks for the info!