  In this paper, for each $d>0$, we study the minimum integer $h_{3,2d}\in
\mathbb{N}$ for which there exists a complex polarized K3 surface $(X,H)$ of
degree $H^2=2d$ and Picard number $\rho (X):=\textrm{rank } \textrm{Pic } X =
h_{3,2d}$ admitting an automorphism of order $3$. We show that $h_{3,2}\in\{
4,6\}$ and $h_{3,2d}=2$ for $d>1$. Analogously, we study the minimum integer
$h^*_{3,2d}\in \mathbb{N}$ for which there exists a complex polarized K3
surface $(X,H)$ as above plus the extra condition that the automorphism acts as
the identity on the Picard lattice of $X$. We show that $h^*_{3,2d}$ is equal
to $2$ if $d>1$ and equal to $6$ if $d=1$. We provide explicit examples of K3
surfaces defined over $\mathbb{Q}$ realizing these bounds.
