I want to show that for two non-zero non-parallel vectors $\vec a$ and $\vec b$, if $$x\vec{a} + y \vec b = 0$$ then $x = y = 0$.
This seems quite obvious but I cannot find an obvious way to prove it. I tried using a contrapositive proof instead.
Suppose if $x \neq y \neq 0$. Then since components of the two vectors are also non-zero, then their resulant vector is also non-zero, thus proving the contrapositive. However, this seems too informal and I don't have a way to show the linear combination of 4 non-zero numbers (from $\vec a$, $x$, $\vec b$, $y$) will certainly give something non-zero as well, since they might as well cancel out.
This claim is not true.
For example, take $\vec{a}$ any vector and take $\vec{b}=\vec{a}$ (since any vector is clearly parallel to itself), then $x=1,y=-1$ satisfies the equation.
Edit: You're on the right track. What you want to show is that if $x\vec{a}+y\vec{b}=0$, and either $x \neq 0$ or $y \neq 0$, then $\vec{a}$ and $\vec{b}$ must be parallel. Think about how their being parallel will relate to their components.