When would a parallel circuit work better than a series circuit?

Efficiency isn't the question. They just do different things. For example, if you look at a typical lighting circuit in a house, all the bulbs on a circuit are in parallel. They each have the same voltage across the bulbs.

If you rewired this so that the bulbs were in series then the voltage would be divided across each bulb and if one bulb burned out the others in series would get no current and would not light.

If you had two 60Watt bulbs in parallel in your house, each would draw 1/2 Amp. Add another bulb and it would draw 1/2 amp as well, for a total of three 60W bulbs drawing 1.5 amps from the power source. If the same three bulbs were in series there would be 40 Volts across each one with a current of 1/6 amps per bulb. Hence each bulb would be about 1/3 as bright as in the parallel circuit.