  Given a K3 surface $X$ over a number field $K$ with potentially good
reduction everywhere, we prove that the set of primes of $K$ where the
geometric Picard rank jumps is infinite. As a corollary, we prove that either
$X_{\overline{K}}$ has infinitely many rational curves or $X$ has infinitely
many unirational specializations. Our result on Picard ranks is a special case
of more general results on exceptional classes for K3 type motives associated
to GSpin Shimura varieties. These general results have several other
applications. For instance, we prove that an abelian surface over a number
field $K$ with potentially good reduction everywhere is isogenous to a product
of elliptic curves modulo infinitely many primes of $K$.
