I am trying to determine if a point $P$ is in a parallelogram.
The parallelogram is defined by $3$ points $A, B, C$.
Point $P$ lies on the same plane as the parallelogram.
I am trying to use fact if $s$ and $t$ are in $[0,1]$ then $P$ is in the parallelogram but I do not know how to solve, $AP = s \cdot AB + t \cdot AC$ for $s$ and $t$.
As I said you in my previous comment, the identity should be:
$$\vec{AP} = s\cdot \vec{AB} + t\cdot \vec{AC}$$
Written under the form $P-A=s(B-A)+t(C-A)$, it gives rise to this linear system with the two first equations with $x$ and $y$ coordinates (the third one with $z$ coordinates being redundant if $P$ is in the plane defined by $A,B,C$);
$$\begin{cases}P_x-A_x=s (B_x-A_x)+t(C_x-A_x)\\ P_y-A_y=s (B_y-A_y)+t(C_y-A_y) \end{cases}$$
Solve it by using Cramer's formulas for $s$ and $t$, then test whether $0 \le s,t \le 1$.