The current would encounter the resistance twice, and the double resistance would halve the current, so that the bulbs would glow only feebly. Connecting the bulbs in parallel means that electricity goes through each bulb only once.
If the wrong-sized wire is used, or if a wire becomes broken or disconnected, this can cause an accidental short circuit which bypasses the resistance of the component. The current in the wire may become dangerously high and melt the wire or cause a fire.
To guard against this, ancillary circuits have fuses. The most common type of fuse is a short length of thin wire enclosed in a heatproof casing often glass. The size of the fuse wire is the thinnest that can carry the normal current of the circuit without overheating, and it is rated in amps.
Most car handbooks and service manuals include a wiring diagram which can be difficult to follow. The colour-coding, however, is a useful guide to tracing wiring.