Parsec is a portmanteau of the words Parallax and arcsecond; an object one parsec distant has a par allax of one arc sec ond
The definition for a parsec being ~3.26 light-years is essentially trigonometric in origin. If observed from Earth an object appears to move a distance of one arcsecond in the sky when is at two opposing points in its orbit (say June and December six months apart) then it must be a certain distance away.
In the diagram above for a parallax angle p of one arcsecond (so ##4.85*10^-6 rad##) we can apply some simple trigonometry to derive the value of one parsec.
The angle ##p=4.85*10^-6 rad## and the length of the shortest edge of the right triangle formed in the diagram (Sun-Earth line) is ##1AU~=1.5*10^11m##.
So the length of the edge connecting to the nearby star ##x## is given by rearranging ##tan(4.86*10^-6 rad)=(1.5*10^11m)/x##
Rearranged for ##x##: ##x=(1.5*10^11m)/(tan(4.85*10^-6 rad))=3.093*10^16m##
This distance happens to be 3.26 light years or so and as such a new unit was defined to make measuring these distances easier; the parsec.