Electricity takes the path of least resistance?

It's not true. To see this, you can try an experiment with some batteries and light bulbs. Hook up two bulbs of different wattages (that is, with different resistances) in parallel with a single battery:

 ------------------------------------------
|                     |                   |
Battery              Bulb 1              Bulb 2
|                     |                   |
 ------------------------------------------

Both bulbs will light up, although with different brightnesses. That is, current is flowing through the one with more resistance as well as through the one with less resistance.

No. The statement is not correct. Current will take any path that is available to it. Which means it can even take the path of leaking out of the wire into the surrounding air, which is seen as sparks when dielectric breakdown of air happens. What you intend to mean perhaps is why current distributes itself in the inverse ratio of resistances, given the same potential difference across different resistive elements.

Ohm's law $I=\frac{V}{R}$ would explain what you are asking. Given a common potential, the amount of current flowing through a resistive element is inversely proportional to the resistance. This would mean, and hopefully answer your question, that a path of lower resistance will have more current flowing through it and vice versa. (Normally the resistance of air is so high that current taking that path and leaking out of the cable is negligibly zero under normal circumstances.)

For a more thorough explanation, currents (and voltages) are distributed to minimize the total power dissipated as heat. This is a consequence of making the action of a disspative system stationary

$\int_{t_1}^{t_2}(L+W)dt$

Here W is the virtual work done by dissipative elements (resistance, capacitance, inductance etc) and L is the dissipation free dynamical system

For an alternative, this link explains how Ohm's Law corresponds to Fermat's Principle of least time.

"Least resistance" can be interpreted as least heat generation. There might be such principle, at least I can show it for @Ted Bunn example so that the answer would be "yes". The largest difficulty in formulating extremal principles is specifying the constraints. I chose fixed current, because I don't see a way to fix voltage for the model in hand without fixing everything else.

In any case I think reformulating least resistance as least dissipation under certain constraints is a right direction.

---

What you have is two bulbs connected in parallel. Let's fix the overall current $I$ through the bulbs rather than voltage $U$. That is it is a case when you are to push a certain amount of electricity through the system. In this setting currents on the bulbs $I_1$ and $I_2$ would be to minimize heat dissipation:

$$ \begin{cases} I_1 + I_2 = I, \\ I_1^2 R_1 + I_2^2 R_2 \to \min \end{cases} $$

Using Lagrange multipliers:

$$ \begin{cases} I_1 + I_2 = I, \\ d \left[ I_1^2 R_1 + I_2^2 R_2 + \lambda (I_1 + I_2 - I) \right] = 0 \end{cases} $$

which leads to

$$I_1 R_1 - I_2 R_2 = 0$$

Thus having assumed the extremality of current distribution we arrived to the distribution which is in harmony with the Ohm's law. One can check that it correspond to the minimum of heat dissipation.