Let $S$ be a triangle with vertices at A=$(1,2,0)$, B=$(2,3,2)$, and C=$(0,0,4)$. Find a parametrization of the triangle and its contour.
I have set up the parametric equations for each edge
$\begin{pmatrix} 2 \\ 3 \\ 2\end{pmatrix}$-$\begin{pmatrix} 1 \\ 2 \\ 0\end{pmatrix}$=$\begin{pmatrix} 1 \\ 1 \\ 2\end{pmatrix}$
first edge: $r_1(t)=\begin{pmatrix} 1 \\ 2 \\ 0\end{pmatrix} + t\cdot \begin{pmatrix} 1 \\ 1 \\ 2\end{pmatrix}=\begin{pmatrix} 1+t \\ 2+t \\ 0+2t\end{pmatrix}$
likewise for the others
$r_2(t)=\begin{pmatrix} 2-2t \\ 3-3t \\ 2+2t\end{pmatrix}$
$r_3(t)=\begin{pmatrix} 0+t \\ 0+2t \\ 0-4t\end{pmatrix}$
for each $t \in [0,1]$
What can I do now? How to put them together, and how to get the parametrization of the whole surface and not only the edges?
What you have done is all you need to do for the edges. You could parametrize so you could go from $0 \leq t \leq 1, 2 \leq t \leq 2, 2 \leq t \leq 3$.
$r_1(t)= (1, 2, 0) + (1, 1, 2)t = (1 + t, 2 + t, 2t) \,$ for $( 0 \leq t \leq 1)$
$r_2(t)= (2, 3, 2) + (-2, -3, 2) (t-1) = (4-2t, 6-3t, 2t) \,$ for $( 1 \leq t \leq 2)$
$r_3(t)= (0, 0, 4) + (1, 2, -4) (t-2) = (-2 + t, -4 + 2t, 12-4t) \,$ for $( 2 \leq t \leq 3)$
Now if you have to parametrize the surface, find the equation of the plane it is on. First take two directional vectors you found above and do a cross product to find the normal vector to the plane-
$(-2, -3, 2) \times (1, 1, 2) = (-8, 6, 1)$
Take any point on the surface say $(0, 0, 4)$ then we have,
$ -8(x-0) + 6(y-0) + (z-4) = 0$
Parametrization of the surface of the plane the given triangle is on,
$ \,z = 4 + 8x - 6y$
EDIT:
On the parametrization that you asked,
If you take parametrization of two of your edges, $r_1(s) = (1 + s, 2 + s, 2s) \,(0 \leq s \leq 1)$
$r_2(t) = (2 - 2t, 3 - 3t, 2 + 2t) \,(0 \leq t \leq 1)$
You can combine these two to parametrize your triangle.
$r(s, t) = (1 + s - 2t, 2 + s - 3t, 2s + 2t) \, \, (0 \leq s, t \leq 1)$
If you take cross product $r_t \times r_s$, you will get the normal vector $(-8, 6, 1)$ which is what I used for other parametrization.