Weierstrass Equation and K3 Surfaces Let $a_{i}(t) \in \mathbb{Z}[t]$. We shall denote these by $a_{i}$. The equation $y^{2} + a_{1}xy + a_{3}y = x^{3} + a_{2}x^{2} + a_{4}x + a_{6}$ is the affine equation for the Weierstrass form of a family of elliptic curves. Under what conditions does this represent a K3 surface?

A good reference for this would be Abhinav Kumar's PhD thesis, which you can find here. In particular, look at Chapter 5, and Section 5.1. If an elliptic surface $y^2+a_1(t)xy+a_3(t)y = x^3+a_2(t)x^2+a_4(t)x+a_6(t)$ is K3, then the degree of $a_i(t)$ must be $\leq 2i$.
I hope this helps.
