  We prove the $W^{1,2}_{p}$-solvability of second order parabolic equations in
nondivergence form in the whole space for $p\in (1,\infty)$. The leading
coefficients are assumed to be measurable in one spatial direction and have
vanishing mean oscillation (VMO) in the orthogonal directions and the time
variable in each small parabolic cylinder with the direction depending on the
cylinder. This extends a recent result by Krylov [17] for elliptic equations
and removes the restriction that $p>2$.
