Well, you can't just see it in terms of the singular K3 but you can see it as the kernel of the specialization map $H_{2}(X_{t}) \to H_{2}(X_{0})$. This specialization map is induced from the retraction of the tubular neighborhood of $X_{0}$ onto $X_{0}$.

It is Theorem 14.1 in Chapter II. Incidentally, the characteristic zero case was originally proven by Serre in his paper on proalgebraic groups: archive.numdam.org/article/PMIHES_1960__7__5_0.pdf