I don't understand why we represent functions $f:I \subseteq \Bbb R \to \Bbb R^2$ the way we do. I don't understand why we represent functions $f:I \subseteq \Bbb R \to \Bbb R^2$ the way we do: doing an analogy with how we represent functions from $\Bbb R$ to $\Bbb R$ or from $\Bbb R^2 \to \Bbb R$, my impression would be to represent functions like $f$ in the format: $$(t,f_1(t),f_2(t))$$
But instead we grab all the points $(f_1(t),f_2(t))$, that is, the image of $f$ and we plot that in $\Bbb R^2$, and we call $f$ a parametrization. 
As an example of this, let $f(t)=(\cos t, \sin t)$. If we represented this by plotting the points $(t,\cos t, \sin t)$ we'd get a helix. But instead we 'splash' that helix into the wall and we get the parametrization of a circle.

We do not "represent" functions $f:I \subseteq \Bbb R \to \Bbb R^2$ as $(f_1(t),f_2(t))$.
The functions $f:I \subseteq \Bbb R \to \Bbb R^2$ are represented as $(t,f_1(t),f_2(t))$ (where $t$ varies on some subset of real line) just like other kinds of functions.
$(f_1(t),f_2(t))$ (where $t$ varies on some subset of real line) is the set which is called the image of function $f$.
Image of a function is an important concept, but "image of a function" ($(f_1(t),f_2(t))$ over varying $t$) and "graph of a function" ($(t,f_1(t),f_2(t))$ over varying $t$) are two different concepts. You should just understand that both these two: a. exist; b. are important; c. are different.
