  We analyze the Brauer-Manin obstruction to rational points on the K3 surfaces
over $\mathbb{Q}$ given by double covers of $\mathbb{P}^2$ ramified over a
diagonal sextic. After finding an explicit set of generators for the geometric
Picard group of such a surface, we find two types of infinite families of
counterexamples to the Hasse principle explained by the algebraic Brauer-Manin
obstruction. The first type of obstruction comes from a quaternion algebra, and
the second type comes from a 3-torsion element of the Brauer group, which gives
an affirmative answer to a question asked by Ieronymou and Skorobogatov.
