How to argue that the potential is the same in a conductor with zero resistance when we have electricity? I am wondering how we can argue that the electrical potential is the same in point A and B in the figure below:

I have seen the argument that in a conductor where the charges are not moving, then the field inside the conductor must be zero, or else the charges would be moving. Also the field at the surface must have zero tangential line, or else you could form a closed loop that could give the electron energy, and this contradicts the conservative property of electrostatic fields. And since the field is zero, the line integral which gives the potential difference is zero.
But now we are in a situation where the charges are moving(we have a current). So how can we argue that the potential difference between A and B is zero? My only idea is that if the potential difference was not zero, there would be some way that charge would be built up at either A or B.
But how do we do this rigorously? Intuitively I understand that if the potential, that is, the line integral of the field is not zero, then that field would move the charges, but how do we show this? What would go wrong if A and B had different electrical potential?
PS:
The answer to this question should be at a more fundamental level than that the potential difference over parallel resistors is the same. This property seems to be used to prove that the voltage drop over parellell resistors is the same.
So I think this question does not have so much to do with the parallell resistors. It could just as well have been a circuit with series resistors, and we looked at two points that were not separated by a resistor.
EDIT:
I got very good answers from different people so it is hard to accept just one. From what I got from different sources I think the answer is this, I'll post it if anyone is interested(this is just my interpretation, others may be better):
If there is a potential difference then the electric field will do work on an electron that moves from A to B. Since potential difference is a line integral and line integrals can be approximated by Riemann sums, we have that across the path from A to B the electrical field will drive electrons forward on small $\Delta r$ intervals(r is path from A to B). We can use Ohms law on these small $\Delta r$ segments and use that $R=0$ and we will have instantaneous and fast movements of electrons. When electrons move we will fast get a new equilibrium where the field must be zero.
 A: 
So how can we argue that the potential difference between A and B is
zero?

You need to start with the definition of potential difference (voltage).
The potential difference $V$ between two points is the work per unit charge required to move the charge between the two points.
If there is no resistance to overcome between two points, no work is required to move charge between the two points.
The mechanical analogy is the work required to move an object between two points on a perfectly frictionless surface. Since a frictionless surface offers no resistance to the movement of the object, no work is required to overcome friction and move the object between the two points.
The thing is, there is no such thing as a perfectly frictionless surface, just like there is no such thing as zero resistance in a wire (with the exception of super conductors). So in the analysis of circuits like yours the resistance between points A and B can be considered to be so low compared the the resistance of circuit components R1 and R2 so that it can be neglected and therefore the potential difference can be considered to be zero.
Hope this helps.
A: It's a convention in electrical problems that ideal wires are assumed to have low enough resistance that we can treat them as equipotential regions.
If we wanted to study a circuit that used very thin wire, or wire made of a material with high resistivity (for example, steel wire rather than copper), or where very small differences in potential are important, we would represent that by drawing additional resistors in the circuit to represent the resistive effects of those wires.
A: 
I have seen the argument that in a conductor where the charges are not moving, then the field inside the conductor must be zero

Fields create forces on charges.  So a non-zero field will accelerate charges.  A static charge implies zero acceleration.

But now we are in a situation where the charges are moving(we have a current). So how can we argue that the potential difference between A and B is zero?

At steady-state (and only then), the field inside the wire must be exactly sufficient to overcome any resistance losses in that segment.  If we assume zero resistance, then there will be zero resistance loss and zero electrical field (again assuming the current is constant).  Once again the zero acceleration ($dI = 0$) is satisfied with zero net force on the charges.
If A and B had different potential, then charges between them would accelerate.  This acceleration would change the number of charges at A with respect to B.  These relocated charges would create their own contribution to the electric field.
Assuming there are sufficient mobile charges (as there are in a conductor), this continues until the net electric field between A and B becomes zero.  At that point charges stop accelerating.
