Let $v_1$ and $v_2$ be given vectors. $v = t_1v_1 + t_2v_2$ varies over the plane determined by the two vectors. The plane is parameterized by $t_1$ and $t_2$. Let $v_0$ be another given vector. What does $v = v_0 + t_1v_1 + t_2v_2$ parameterize?
I'd say that would be a plane since $v_0$ is a single non-scaled vector. Is that right?
Imagine the whole plane being moved by the vector $v_0$. What then do you get?