I will consider only the case $d=2$ (and for simplicity $K=\mathbb{Q}$ if you wish).

As Silverman says in his answer and his comment, there are two distinct but related problems: one is to consider the points of the curve $Y$ (that I assume smooth and projective of genus $\ge 2$) defined on some quadratic extension of $K$, and the other the points defined over $K_2$, the compositum of all quadratic extensions of $K$. The first case is already answered in the paper by Harris and Silverman cited above (the answer being the hyperelliptic curves and the bielliptic curves with an elliptic quotient with infinitely many points over $K$, so their gonality is bounded by $4$), so I will say same words on the second case.

First, the field $K_2$ can be written as the union of all fields $L/K$ which are abelian extensions with Galois group $(\mathbb{Z}/2\mathbb{Z})^r$ for some $r\ge 1$. So you are considering now which curves have infinitely many points defined over this fields. Since there are much more fields you expect to have other examples than the ones above.

For example, all curves that have a map $\pi$ to $\mathbb{P}^1_K$ which is Galois with Galois group isomorphic to $(\mathbb{Z}/2\mathbb{Z})^r$ for some $r\ge 1$ (with this I mean that the corresponding extension between the function fields is Galois, etc, but not that the map is unramified). This curves can have gonality as big as you wish, so you get much more examples.

To be more explicit, you can consider $f_1(x),\dots, f_r(x)\in K[x]$ pairwise coprime polynomials of degree $\ge 3$, and the curve $Y$ to be the projectivization of the affine curve in the affine space $\mathbb{A}_K^{r+1}$ given by the equations $y_i^2=f_i(x)$ for $i=1,\dots, r$. It is clear that this curve has infinitely many points in $K_2$ just taking the points with coordinate $x\in K$. On the other hand this curve has (at least generically, but I think always under the given conditions) gonality $2^r$, with gonal map given by sending a point $(x,y_1,\dots,y_r)$ to $x$. More general cases with not necessarily pairwise coprime polynomials, or with degrees $\le 2$, or with terms of degree $1$ in the $y_i$ can be taken, but the gonality is not so easy to control.

Of course you can consider also the curves $Y$ with a map $\pi:Y\to E$ to an elliptic curve $E$ defined over $K$, with $\sharp E(K)=\infty$ and which is Galois with Galois group isomorphic to $(\mathbb{Z}/2\mathbb{Z})^r$ for some $r\ge 1$. I don't know if there other other examples of curves that these two types.

Finally, you talk in the message of plane curves, but I assume you are considering plane singular models of the curves. If you really only want to consider curves with non-singular plane projective models, then one should study if these type of curves but with high degree can have a map $\pi$ as described above.