Is $\{ 0^p1^q0^r \mid p \neq r \}$ context-free?
The mistake in your proof is that the pumping lemma says that such a division (w=uvxyz) exists.
So in order to say $L_2$ is not context free, you have to show that for every division, you get a contradiction.

In your example we can use this division:
$u=01, \, v=00, \, x=\epsilon, \, y=\epsilon , z=\epsilon$
So for every $n \geq 0$:
$uv^nxy^nz=01(00)^n \in L_2$