Voltage in parallel and series connection

In series connection, voltage is said to be different and current constant across the circuit. How does it come about ? Is it due to the accumulation of electrons on one side due to resistors? Or, if it is so, how is the situation reversed in the case of parallel connection?

In the series connection the current (charge passing a point per second) is the same every where because there are assumed to be no sources or sinks of charge and so the charge is conserved - the amount of charge entering is equal to the amount of charge leaving. This is Kirchhoff's current law and there is certainly no accumulation of charge rather the charges are pushed through the resistors by the electric field produced by the battery.

You can liken the situation to some pipes connected in series with a pump forcing water through the pipes.
The rate at which water flows through the pipes (the current) is the same for each pipe although the pressure difference (the voltage) across each of the pipes is different.

Voltage is to do with the conversion of electrical energy into other forms of energy and the potential difference between two points is the work done in taking unit positive charge between those two points. The work is done by the electric field produced by a power supply.

If you have a 10 ohm and a 5 ohm resistor in series and a current of $3$ amps going through them both then the voltage across the $10$ ohm resistor is $3 \times 10 = 30$ volts and the voltage across the $5$ ohm resistor is $5 \times 3 = 15$ volts.
What this means is that in taking one coulomb of charge through the $10$ ohm resistor $30$ joules of electrical energy is converted into heat whereas in taking the same amount of charge through the $5$ ohm resistor produces $15$ joules of heat.

In the case of resistors in parallel then the current which flows through each resistor is different but the work done in taking unit positive charge through each resistor, the voltage across the resistors, is the same.
This idea of the same work being done in taking unit charge between two points along different routes is very important.

This is equivalent to having a number of pipes all connected to the same outlet from a pump and the other end all connected together so the pressure exerted difference on the water (the voltage) across each of the pipes is the same although the rate of flow of water (current) through the pipes is different.

So with a $10$ ohm and a $5$ ohm resistor in parallel and a voltage of $10$ volts across them the current though the $10$ ohm resistor is $\frac {10}{10} = 1$ amp and the $5$ ohm resistor is $\frac {10}{5} = 2$ amp
More current flows through the path of least resistance ie through the $5$ ohm resistor but the conversion of electrical energy into heat per unit charge passing through eother resistor is the same.