In this paper we study, for each $d>0$, what is 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 $\rho (X):=\mathrm{rank}\, \mathrm{Pic}\, X = h_{3,2d}$ admitting an automorphism of order $3$. We show that $h_{3,2d}=6$ if $d=1$ and $h_{3,2d}=2$ if $d>1$. We provide explicit examples of K3 surfaces defined over $\mathbb{Q}$ realizing these bounds.