Rational points on K3 surfaces of degree 2

A K3 surface over a number field has infinitely many rational points over a finite field extension. For K3 surfaces of degree 2, arising as double covers of $\mathbb{P}^2$ branched along a smooth sextic curve, we give a bound for the degree of such an extension. Moreover, using ideas of van Luijk and a surface constructed by Elsenhans and Jahnel, we give an explicit family of K3 surfaces of degree 2 defined over $\mathbb{Q}$ with geometric Picard number 1 and infinitely many $\mathbb{Q}$-rational points that is Zariski dense in the moduli space of K3 surfaces of degree 2.