I read here (on page 3) that
"Every plane parametric curve (or surface) can be expressed as an implicit curve (or surface). Some, but not all implicit curves (or surfaces) can be expressed as parametric curves (or surfaces)".
It also states that this fact comes from Algebraic Geometry.
I was interested in the parameterisation of continuous curves (or surfaces). Can every continuous curve (or surface) be parameterised? If not, then under what conditions they can't be?
There are obstructions to algebraic curves (and more general varieties) being parametrizable. I will discuss the case of projective curves for simplicity, but all of the main ideas extend to higher dimension. Let us even assume we have a smooth projective curve $C$.
A parameterization for $C$ amounts to a surjective morphism $\phi : \mathbb P^1 \to C$. By the Riemann--Hurwitz formula, this forces the genus of $C$ to be $0$ (and in fact forces $\phi$ to be an isomorphism). In the case of smooth curves over the complex numbers, the geometric, arithmetic, and topological notions of genus all agree, so you can think of this as being the "number of holes" appearing in the compact Riemann surface associated to a smooth projective curve over $\mathbb C$. When the curve is not smooth, more care has to be taken, but I'll ignore those subtleties here.
So a good class of examples of curves that cannot be parametrized are smooth curves of nonzero genus, e.g. elliptic curves. The example given in Jan-Magnus's comment above is an elliptic curve. These are the smooth projective curves of genus $1$. In characteristic $>3$, these are precisely the curves of the form $y^2=x^3+ax+b$ where the RHS does not have a double root. Curves of higher genus will also give examples.