No, they do not all have the same voltage drop. If they were in series, however, they would.
By Ohm's Law, the voltage drop is proportional to the current flowing through a resistor. (So in several series resistors with the same resistance, the drop across each one is the same, since the current across each one is the same). However, because B and C are in parallel, the current is split between those two, meaning that the current in B and C is different from that passing through A.
However, since B and C have the same resistance, you know that the current from A is split 50/50 between them, so since each one gets half of the current, the voltage drop across each one is half of the voltage drop across A.
To find the current, first find the equivalent resistance using parallel and series resistor simplification techniques. Condense the two parallel 10 ohm resistors into a single resistor, and then combine this resistor with the series 10 ohm resistor. This tells you the total (or "equivalent") resistance of the circuit. Since you know the voltage produced and the resistance of the circuit, you can use Ohm's Law to find current.
The key thing is that there is NO electric field within the perfect wire. So, there is no force acting on the electron, and thus no work done on it (while it's in the perfect wire).
This goes back to the definition of a perfect conductor (which the perfect wire is). Within a perfect conductor, there is no electric field. Instead, the charges (which have infinite mobility) rearrange themselves on the surfaces of the conductor in such a way as to perfectly cancel out any internal field.
So, the only fields in your circuit would be 1) in the battery, and 2) in the resistor.
I should also add that this is due to the approximation of the wire as 'perfect'. A real wire has some resistance, or equivalently, its charges don't perfectly reorder so as to perfectly cancel an internal field.
Best Answer
Consider a voltage source, e.g. an ideal 5V cell phone power supply. There is a 5V drop between Source and Ground.
Now put in a resistor of let's say 10kOhm between Source and Ground. The voltage drop across this resistor is 5V.
Now wire in another resistor of 10kOhm - parallel or serial does not matter for this part.
What is the voltage drop across the system of resistors? It is 5V - because that is what the source supplies.
If they are in series, it is 2.5V over the first, and then 2.5V over the second, for a total drop of 5V.
If they are in parallel, it is 5V over the both of them because the source creates a 5V voltage across the both of them. Wire in any number of resistors in parallel and the voltage across them will stay 5V because "The voltage across them" is the voltage of the source, which is 5V.