A parametrization of an object (such as a line or a plane) in $\mathbb{R}^n$ usually refers to an injective (i.e., one-to-one) function $\phi$ from $\mathbb{R}^k$ (or a subset of $\mathbb{R}^k$) to $\mathbb{R}^n$ that is a bijection onto its image (usually that bijection is in fact a homeomorphism or diffeomorphism, depending on the context).
Your first example is the function $\phi_1: \mathbb{R} \to \mathbb{R}^2$ given by $\phi_1(t) = (t, 2t)$. Crucially, $\phi_1$ is a bijection from $\mathbb{R}$ onto its image, which is a line in $\mathbb{R}^2$.
Your second example is the function $\phi_2: \mathbb{R}^2 \to \mathbb{R}^2$ given by $\phi_2(t, k) = (t + k, 2(t+k))$. Although $\phi_2$ is a surjective function onto the line in question, it's not injective since multiple points in the domain map to the same point in the image (e.g., $\phi_2(0,1) = \phi_2(1,0) = (1,2)$). This means it's not a "parametrization" in the usual sense. You should think that because $\phi_2$ is not injective, one of the two variables is "redundant."
Essentially, subsets of $\mathbb{R}^n$ that can be (locally) parametrized in the above sense are called manifolds. Examples of manifolds are lines, planes, and more curvy objects like spheres. There may be many different ways to parametrize a manifold, but, as you suspected, every parametrization must always have the same number of variables $k$. $k$ is called the dimension of the manifold. For example, a line has dimension 1 (as we saw above, using more than one variable to try to parametrize a line introduces a redundancy). Planes and spheres in $\mathbb{R}^3$ both have dimension 2.
