Zero pressure difference under upward acceleration

Answer: the pressure difference becomes zero only when the whole water body has a downward acceleration equal to $g$ (free fall), i.e. $a_z=-g$. For any upward acceleration it can never be zero; it only increases.

Take a fluid element in a body moving with vertical acceleration $a_z$ (upward positive). Newton's second law for a column of height $h$ and unit area:

$$p_2A-p_1A-\rho g h A=\rho h A\,a_z$$

where point 1 is the upper point and point 2 the lower point, a distance $h$ apart. So

$$\Delta p=p_2-p_1=\rho h\,(g+a_z)$$

Setting $\Delta p=0$:

$$g+a_z=0\ \Rightarrow\ a_z=-g$$

For an upward acceleration $a_z>0$, $\Delta p=\rho h(g+a_z)>\rho g h$, so the pressure difference is greater than the static value. At free fall (weightless condition) the fluid has no apparent weight and the pressure is uniform throughout.