I'm trying to understand a statement here. It states that the parametrisation of a cubic function is: $$\begin{align} x(t)&=a_xt^3+b_xt^2+c_xt+d_x\\ y(t)&=a_yt^3+b_yt^2+c_yt+d_y\\ \end{align}$$
Or in compact from $$\begin{align} P(x,y)&=at^3+bt^2+ct+d\\ \end{align}$$
being a, b, c and d 2d vectors.
But I can't get it. How do they get to this conclusion?
Maybe that you are misinterpreting what the link say. The text is:
so these equations are the definition of a parametric cubic curve (in $\mathbb{R}^3$), not the parametric representation of a cubic as: $$y=ax^3+bx^2+cx+d \qquad (1)$$, that is a curve in $\mathbb{R}^2$.
Added after the change in the question.
In $\mathbb{R}^2$ the cubic $(1)$ can be expressed in the parametric form: $$\begin{align} x(t)&=a_xt^3+b_xt^2+c_xt+d_x\\ y(t)&=a_yt^3+b_yt^2+c_yt+d_y\\ \end{align}\qquad (2)$$
simply using $t=x$ and
$a_x=b_x=d_x=0 \quad,\quad c_x=1\quad,\quad a_y=a \quad,\quad b_y=b\quad,\quad c_y=c\quad,\quad d_y=d$
but note that, in general, a cubic expressed as $(2)$ with the coefficients of $t^2$ and $t^3$ not null, cannot be write in the form $(1)$.