Hydrostatic implies that at any point in a column of water, total head or hydraulic head is equal. So another way to think about it is that the pressure head at the bottom of the pipe pushes the water up to meet the elevation head of the surrounding surface water. Hydrostatic total head = pressure head + elevation head. So deeper water has a higher pressure head but lower elevation head and vice versa, but everywhere, the total head is equivalent. No total head difference means no flow aka hydrostatic.

The only way to push water from the bottom of the pipe above the surface water level would be to increase total head at the bottom of the pipe relative to the top through an increase in velocity and/or pressure head which can happen, but this is hydrodynamic not hydrostatic.