Can a circle truly exist? Is a circle more impossible than any other geometrical shape? Is a circle is just an infinitely-sided equilateral parallelogram? Wikipedia says...
A circle is a simple shape of Euclidean geometry consisting of the set of points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius.
A geometric plane would need to have an infinite number of points in order to represent a circle, whereas, say, a square could actually be represented with a finite number of points, in which case any geometric calculations involving circles would involve similarly infinitely precise numbers(pi, for example).
So when someone speaks of a circle as something other than a theory, are they really talking about a [ really big number ]-sided equilateral parallelogram? Or is there some way that they fit an infinite number of points on their geometric plane?

I think it quite depends on how you understand "exist" and how you define $circle$. For example, in terms of Cartesian coordinates, a circle of radius $1$, centering at the origin of the $x$-$y$ plane, can be "defined" as the set 
$$\{(x,y)\in{\mathbb R}^2:x^2+y^2=1\}$$
Then when you ask whether such set exists or not, the answer is yes, according to the axioms for set theory.
The keyword in the excerpt you quote from wiki, I think, is "set"(the set of points in a plane that...). So when you are asking "can a circle exist" you are actually asking "can such set exist".
