Let $X$ be a smooth projective threefold with $h^{0,1}(X) = h^{0,2}(X)=0$ that has a smooth anticanonical section $D$. Then $D$ is necessarily a $K3$ surface. Consider a subgroup $$Pic_X(D) = i^*(Pic(X))$$ of $Pic(D)$, where $i:D \hookrightarrow X$ is the inclusion map. I tried several examples of $X$, including some blow-ups of toric Fano threefolds and found that $Pic_X(D)$ is a primitive subgroup of $Pic(D)$(i.e. the quotient $Pic(D) / Pic_X(D)$ has no torsion)

Is it generally true or are they any counterexamples?

If $X$ is a smooth Fano threefold, then it is obvious but I am not sure about the general cases.