Rune spent many years as a wandering master, teaching mathematics to the children of relatively wealthy families, who could afford his services. It was a subject which most people didn't feel required a specialized master, as parents could teach their children simple math, such as addition, subtraction, multiplication, division, and such. However, some occupations required an understanding of more complex equations, and so anyone wishing to enter such fields would study under masters like Rune.
In the course of his wandering, he often met other masters, both fellow wanderers and those who stayed in one village their whole lives. At some point, he became close friends with a history master named Alphonse, and the two would sometimes travel between villages together, even obtain simultaneous employment by some families. However, in 902, during the Coming of the Order, when the school system was first suggested, Al realized that the days of independent masters was coming to an end, and returned to his former career as a merchant sailor. Rune, on the other hand, preferred to hold out hope that people like himself would continue to find employment. Besides, there was really nothing else he felt qualified to do.
After schools were finally established in 904, Rune looked into them, but decided he wanted no part of them. Failing to find private work, he eventually became a street rat. When the surname law was passed that same year, he took to calling himself "Parallelogram," which had always been his favorite geometric shape (in large part just because he liked the sound of the name). However, soon after that, it became rare for anyone to have reason to address him at all, and if anyone did, they'd likely just call him "Streetrat," a surname which was applied to most homeless people (though it could be unclear whether people addressed such folks by a surname, or were simply calling them "street rat").
In 912, Rune met Darius Lonewander and Tiejo Streetrat, two of the founding members of The Chaos. He went with them to Shanty, when Darius first went to find pirates who they might convince to organize a navy for them, as part of their plans of rebellion against the Second Order. The pirate captain they contacted was the commander of a ship called Woodsorrow, on the crew of which Darius's cousin Cara MakeDo served as a junior officer (having joined the crew just a year earlier). By an amazing coincidence, Cara's captain just so happened to be Alphonse Teach, Rune's old friend.
It's unknown how much of a part, if any, Rune played in convincing his old friend to help Lonewander, nor whether Rune himself played any part in the ensuing Chaos War, later that year. However, it is known that he later helped Tiejo organize the Allrat Fellowship in late 912, and when that organization divided itself into separate "guilds" for each type of 'Rat, Rune became the guildmaster of the Street rats' guild. (As a side note, while Tiejo sometimes refers to himself as "Tiejo Fellowship Streetrat," Rune is sometimes called "Rune Parallelogram Streetrat," following Tiejo's example of choosing an unofficial middle name; though in Rune's case, it's more like shifting his original surname to a middle position.)