Prove that $S_2$ is closed under union and complement
As you mention in your post, you can define a new verifier $V=V_A \vee V_B$ however, you missed the point that the input of $V$ need not be the input of $V_A$ or $V_B$: it can be the concatenation of their inputs

So the new $V$ can be written as
$$ V(x, y_A\circ y_B, z_A \circ z_B )$$
where it "runs" $V_A(x,y_A,z_A)$ and $V_A(x,y_B,z_B)$ and decides accordingly.

It then immediately follows that $q = p_a +p_b +1$  (the +1 is just to separate the prefix from the suffix, you can ignore it).