I believe that if we were just dealing with electrons, the repulsion from the electric field would be much much stronger than the attraction from the magnetic field.
We only see a net attraction between current carrying parallel wires because the wires contain both electrons and protons in almost perfect balance, so that the electron/electron repulsions and proton/proton repulsions between the wires are almost perfectly canceled by the electron/proton attractions between the wires, leaving no net electrical attraction or repulsion.
That lets the weak magnetic attractions (or repulsions, depending on the directions of the currents) dominate.