The main advantage of wiring light bulbs in parallel rather than in series is so that when one burns out the other stays lit. Parallel circuits are branched and provide more than one pathway through which electrical current (electrons) can flow. A series circuit has only one pathway so that current must flow through all parts of the circuit. If any part of it is interrupted the circuit is broken and current ceases to flow. An analogy is a road that's blocked by a rock slide. If there's no alternate route traffic stops, but if there is an alternate route, as in a parallel circuit, traffic will flow around it. The image below shows the difference between parallel and series circuits. Current splits at the junction of the two branches and goes through both.
The thin wire filament in a light bulb is part of the circuit. It has very high resistance because it's so thin. This causes the electrons to slow down and heat up the wire, producing heat and light. When the filament breaks the circuit is broken and electrons can no longer move through it.