The reciprocal rule
The parallel rule comes out of the same two facts as the series one — but this time it is the currents that add, and that is what puts everything upside down.
Start from the fact that the branch currents add up to the total, and substitute V ÷ R for each current using Ohm's law. Every branch has the same V across it, and R is the equivalent resistance of the whole arrangement.
I = I1 + I2 + I3
V ÷ R = V ÷ R1 + V ÷ R2 + V ÷ R3
Every term now contains V, so dividing through by it leaves the rule.
Resistors in parallel
1 ÷ R = 1 ÷ R1 + 1 ÷ R2 + 1 ÷ R3
The reciprocal of the equivalent resistance equals the sum of the reciprocals of the individual values.
The answer is not R yet
The formula gives you 1 ÷ R, not R. Turning it upside down at the end is a step that is very easy to forget, and it turns a right method into a wrong answer.
A check you can do in your head
The equivalent resistance of a parallel set is always smaller than the smallest resistor in it. If your answer for two resistors of 12 and 6 ohms is bigger than 6, you have gone wrong somewhere — most likely by forgetting to invert at the end.
Why parallel lowers the resistance
Adding a branch gives the current an extra route it did not have before, so more current flows for the same voltage — and more current for the same voltage is exactly what a smaller resistance means.