Since we know current can not flow in a wire if there is no potential difference in its two terminals. How current flows in a parallel combination where each load has equal potential difference in its two terminals?
In a parallel combination each circuit has the same voltage $U$, which can be easily shown with Kirchhoff's laws. But the potential difference between the two knots of the parallel wired part is nonzero. Therefore current flows, depending on the resistance (where the voltage drops) of the parallel-connected circuit (so that the voltage $U$ is the same in each circuit).
Current does not flow from one point to another point if there in zero potential difference that means if potential difference of both points is equal. As you have said, in parallel combination each load has equal potential difference but the potential difference is not zero at all. Each load has potential difference between two terminals. So current flows from terminal at higher potential to terminal at lower potential. But the amount of current that flows in each parallel wire depends upon the resistance of each wire.