The attractive force between parallel currents is the magnetic force. The magnetic force exists so that the electric force comports with special relativity: if the electric force existed on its own, then the force on a particle wouldn't transform right from one inertial frame to the other. So now the question becomes why does the electric force exist? The electric force is due to the exchange of photons between charged particles.
P.S. We can go further down the rabbit hole. Why is there a coupling between charged particles and photons in the first place? At some point in physics it's safer to say "Beats me."