How was it determined that a parsec is $3.26$ light years?

Parallax is an effective way to measure distance to nearby stars because it relies on geometry. When astronomers use parallax, they are measuring how a star appears to move against its background. The unit parsec refers to the distance that an object would have to be from the Earth  to have a parallax angle of $1$ arc-second.

In the figure above, the angle, $\alpha$ is the angle that is measured by the Earth on opposite sides of the sun. The parallax angle, $p$ is half of this angle.

$p=\frac{1}{2}\alpha$

If we defines $p$ to be $1$ arc-second, then our object will be $1$ parsec away. Since light from the sun takes $8$ minutes and $20$ seconds to reach the Earth,. we know that;

$1AU=8.33$ light minutes

We can use this information to convert our parsec into light years with the tangent formula.

$tan(p)=\frac{8.33lightminutes}{d}$

Or;

$d=\frac{8.33lightminutes}{tan(p)}$

If we convert $p$  to radians, than we can use the small angle approximation, $tan(\theta )\approx \theta$.

$1arc-second=4.85\times 10^{-6}$ radians

Plugging this in for $p$  and using the small angle approximation;

$d=\frac{8.33lightminutes}{4.85\times 10^{-6}}$

=$1.72\times 10^6$ light-minutes

=$3.26$ light years