How could the potential difference be constant across all the resistors of parallel connecting resistors although each resistor has a specific resistance?
Kirchoff's laws tell us that the potential drop across any closed loop in a circuit must be equal to the voltage sources in the loop, from which we conclude that the voltage drop across resistors in parallel must be equal.
Ohm's law states:
From which we conclude that, since $V$ is fixed, if the different resistors have different $R$'s, then the current ($I$) through each must also be different (and obey Ohm's law).
This is because what changes in each resistor is the current passing through and not the voltage difference. One the other hand when resistors are in series they have the same current passing through, but different voltage through each one's nodes.
In essence when resistors are in parallel do not share same current path (i.e wire) but share same voltage. On the other hand when resistors are in series they do share the same current path. Since current $I$, voltage $V$ and resistance $R$ are related (macroscopicaly) by Ohm's law ($V = IR$). The rest follows.
Once i asked my high-school physics teacher, how come resistance is defined as "difficulty" of current pasing through and at the same time, resistors in series have exactly same current passing through.
The answer was that although each individual electron has a "difficulty" passing through a resistor (and dissipates energy/heat etc.) The whole electron cloud (which defines macroscopic current) goes on at same average speed.